March 03, 2026
In this paper we investigate the Cauchy problem for Schrödinger ultrahyperbolic equations with singular (less than continuous) coefficients. We prove \(H^\infty\) well-posedness in the very weak sense under suitable assumptions of the distributional structure of the coefficients and decay on the lower order terms. Consistency is proven with the classical \(H^\infty\)-results when the equation coefficients are smooth.
startsection section1@-3.5ex plus -1ex minus -.2ex2.3ex plus .2exIntroduction Schrödinger type equations play a central role in modelling wave phenomena across a wide range of physical contexts, including optics, quantum mechanics, and complex fluid dynamics. This paper is devoted to a specific class of these equations, known as ultrahyperbolic Schrödinger equations, which are characterised by direction-dependent dispersion. This anisotropic dispersive structure makes their mathematical analysis especially subtle and demanding. Their analysis provides insight into how waves and signals propagate, interact, and evolve over time, particularly in media where dispersive effects and material properties exhibit intricate behaviour. Such equations naturally emerge in the study of nonlinear wave phenomena, including multidimensional water-wave models (see [1], [2]) and certain completely integrable systems (see [3]). In detail, we will study the Cauchy problem \[\begin{cases}\label{mainprobIntro} Pu=f \quad \text{in} \;(0,T] \times \mathbb{R}^n, \\ u(0,\cdot)=u_0 \quad \text{in} \;\mathbb{R}^n, \end{cases}\tag{1}\] where \(u_0 \in \mathscr{D}'(\mathbb{R}^n)\), \(f \in C([0,T];\mathscr{D}'(\mathbb{R}^n))\) and \[\label{eq46P} P=D_t-\sum_{i,j=1}^nD_{x_i}(a_{ij}(x)D_{x_j})- \sum_{k=1}^nb_k(x)D_{x_k}-V(x),\tag{2}\] with \(a_{ij},b_k,V \in \mathscr{D}'(\mathbb{R}^n)\), for all \(i,j,k=1,\dots,n\), non-smooth coefficients that satisfy suitable assumptions. Throughout the paper we will use the notation \(D_t=-i\partial_t\) and by \(D_{x_j}=-i\partial_{x_j}\). We will assume that the matrix \(\mathcal{A}=(a_{ij}(x)))_{i,j=1}^n\) is real and symmetric and satisfy a non-degeneracy condition. The term ultrahyperbolic in this context refers to the fact that the operator \(A\) involves different directions in space that have opposing effects on wave propagation. A typical example in \(\mathbb{R}^2\) is given by a diagonal \(2\times 2\)-matrix \(\mathcal{A}\) where \(a_{11}(x)=c_1+\widetilde{a}_{11}(x)\) and \(a_{22}(x)=c_2+\widetilde{a}_{22}(x)\) are suitable real perturbations of constant coefficients \(c_1>0\) and \(c_2\neq 0\). The Cauchy problem 1 has been studied in [4], [5] for space-dependent variable coefficients using the machinery of pseudodifferential calculus, and more recently in [6], where well-posedness in Sobolev spaces is established through a different approach. The methods involved heavily depend on the regularity of the coefficients, mainly continuity in \(t\) and smoothness with respect to the space variable \(x\), and limit the physical applications that might involve discontinuous objects, as jump functions, delta of Dirac, etc.
The aim of this paper is to drop the traditional regularity assumptions in 1 and prove that the corresponding Cauchy problem is well-posed in the very weak sense. The notion of very weak solution has been introduced in the context of hyperbolic equations in [7] to deal with discontinuous coefficients and employed in related work on hyperbolic equations and systems with multiplicities [8]–[12]. The same ideas have been recently applied in the context of Schrödinger type equations with singular coefficients in [13]–[17]. In a nutshell, since the presence of distributional coefficients might not allow a meaningful definition of the operator \(P\), our approach is to replace \(P\) with a family of regularised operators \((P_\varepsilon)_\varepsilon\) where \(\varepsilon\to 0\). This is done via convolution with mollifiers of the type \(\varphi_{\omega(\varepsilon)}(x)=\varepsilon^{-n}\varphi(x/\omega(\varepsilon))\) where \(\omega\) is a regularising net tending to \(0\). Our aim is to identify a suitable regularisation method (mollifier and scale) such that the regularised Cauchy problem \[\begin{cases} P_\varepsilon u=f_\varepsilon\quad \text{in} \;(0,T] \times \mathbb{R}^n, \\ u(0,\cdot)=u_{0,\varepsilon}\quad \text{in} \;\mathbb{R}^n, \end{cases}\] is well-posed and the solution \((u_\varepsilon)_\varepsilon\) fulfills moderate estimates (i.e. \(O(\varepsilon^{-N})\) for some \(N\in\mathbb{N}_0\)) with respect to the parameter \(\varepsilon>0\). This means to prove that a very weak solution \((u_\varepsilon)_\varepsilon\) exists. Its uniqueness is proven modulo negligible nets, i.e., negligible perturbations (i.e. \(O(\varepsilon^{q})\) for all \(q\in\mathbb{N}_0\)) of coefficients and initial data lead to negligible perturbation of the very weak solution. The paper is structured as follows.
Section 2 collects some preliminaries notions on pseudodifferential operators, Sobolev spaces, Sobolev mapping properties and regularisation via convolution with a mollifier. Since Sobolev spaces are the natural environment where to study well-posedness we will work with Sobolev-moderate and negligible nets. Section 3 is devoted to ultrahyperbolic operators with singular coefficients, in particular the regularisation of the principal part \[A=\sum_{i,j=1}^nD_{x_i}(a_{ij}(x)D_{x_j}),\] where \(a_{ij}(x)=c_{ij}+\tilde{a}_{ij}(x)\), \(c_{ij} \in \mathbb{R}\) and \(\tilde{a}_{ij} \in W^{1,\infty}(\mathbb{R}^n)\) is real valued, for all \(i,j=1,\dots,n\). We identify a set of hypotheses on the regularised operator \(A_\varepsilon\) that allow to generalise the Doi’s lemma for ultrahyperbolic operators with regular coefficients to this singular context. This is fundamental step in the proof of our well-posedness result. The full statement of the problem (involving lower order terms as well) and the proof of existence and uniquenss of a \(H^\infty\)-very weak solution are the contents of Section 4 and 5. Our proof method makes use of smoothing estimates as in [4], [18], [19] combined with careful approximation techniques that states clearly the dependence on the regularising scale. We conclude the paper with Section 6 where consistency with the classical result, i.e., when the equation coefficients are smooth then every weak solution will converge to the classical solution as \(\varepsilon\to 0\).
startsection section1@-3.5ex plus -1ex minus -.2ex2.3ex plus .2exPreliminaries Notation. We make use of the notations \(D_t=-i\partial_t\) and \(D_{x_j}=-i\partial_{x_j}\). We write \(\langle \cdot \rangle=(1+|\cdot|^2)^{1/2}\). By \(\mathbb{N}_0\) we mean \(\mathbb{N}\cup \lbrace 0 \rbrace\). We denote by \((\cdot,\cdot)_0\) the \(L^2\) product. For \(s \in \mathbb{R}\) we denote by \(H^s(\mathbb{R}^n)\) the Sobolev space of order \(s\). With \(\langle \cdot, \cdot \rangle\) we denote the standard Euclidean product. The space \(C^\infty_b(\mathbb{R}^n)\) is the space of bounded \(C^\infty\) functions on \(\mathbb{R}^n\), with bounded derivatives at any order.
In this subsection we recall the basics of pseudodifferential calculus we need throughout the work. For the sake of completeness we start with the notion of (standard) symbol in \(\mathbb{R}^{n}\times \mathbb{R}^n\).
Let \(a \in C^\infty(\mathbb{R}^{n}\times \mathbb{R}^n)\) and \(m \in \mathbb{R}\). We say that \(a\) is a symbol of order \(m\), and we write \(a \in S^m(\mathbb{R}^{n}\times \mathbb{R}^n)\) (or simply \(a \in S^m\)), if for all \(\alpha,\beta \in \mathbb{N}_0^{n}\) there exists \(C_{\alpha,\beta}>0\) such that \[\label{eq46symbol} |{\partial_x^\beta\partial_\xi^{\alpha}a(x,\xi)}|\leq C_{\alpha\beta}\langle\xi\rangle^{m-|\alpha|}, \quad (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n.\tag{3}\]
The set of symbols of order \(m\) is a Fréchet space where, if \(a \in S^m\), the semi-norms are given by \[\lvert a\rvert_k^{(m)}:=\max_{\lvert\alpha\rvert+\lvert\beta\rvert\leq k}\sup_{(x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n}\lvert\partial_x^\beta\partial_\xi^\alpha a(x,\xi) \rvert\langle \xi \rangle^{-(m-\lvert\alpha\rvert)}, \quad k \in \mathbb{N}_0.\] Equivalently, note that the Fréchet topology can be given by using the seminorms
\[|a|^{(m)}_{\alpha,\beta}:=\sup_{(x,\xi)\in \mathbb{R}^n\times \mathbb{R}^n}|\partial_\xi^{\alpha}\partial_x^\beta a(x,\xi)|\langle{\xi}\rangle^{-(m-|\alpha|)}, \quad \alpha,\beta \in \mathbb{N}_0^n.\]
If \(a \in S^m\), we can associate to it a pseudodifferential operator using the quantization formula \[\mathrm{Op}(a)u(x):= (2\pi)^{-n}\int_{\mathbb{R}^{2n}}e^{i\langle x-y, \xi \rangle} a(x,\xi)u(y)dyd\xi, \quad u \in \mathscr{S}(\mathbb{R}^n).\]
It is easy to verify that \(\mathrm{Op}(a)\) is a linear operator that acts continuosly on \(\mathscr{S}(\mathbb{R}^n)\) and extends by duality to a linear continuous operator on \(\mathscr{S}'(\mathbb{R}^n)\). If \(A=\mathrm{Op}(a)\), for some \(a \in S^m\), we write \(A \in \Psi^m(\mathbb{R}^n \times \mathbb{R}^n)\) (or simply \(A \in \Psi^m\)).
Furthermore, recall that, if \(a \in C^\infty (\mathbb{R}^n \times \mathbb{R}^n)\) is a real-valued smooth function, \(H_a\) denotes the Hamilton vector field associated with it, which, in standard symplectic coordinates \((x,\xi) \in \mathbb{R}^n\times \mathbb{R}^n\), can be defined as \[\label{eq46defHa} H_a:=\sum_{j=1}^n \Bigl( (\partial_{\xi_j}a)\partial_{x_j}-(\partial_{x_j}a)\partial_{\xi_j} \Bigr)\tag{4}\] and then (with \(b \in C^\infty (\mathbb{R}^n \times \mathbb{R}^n)\)) \[\label{eq46defPoisson} \lbrace a, b \rbrace(x,\xi):=H_ab(x,\xi)=\sum_{j=1}^n \Bigl( \partial_{\xi_j}a\partial_{x_j}b-\partial_{x_j}a\partial_{\xi_j}b \Bigr)(x,\xi) \quad (x,\xi)\in \mathbb{R}^n \times \mathbb{R}^n,\tag{5}\] where \(\lbrace \cdot, \cdot \rbrace\) are known as the Poisson brackets.
Our well-posedness result will be formulated in the context of Sobolev spaces, defined through \(\Lambda^s=\mathrm{Op}(\langle \xi \rangle^s)\), as \[H^s(\mathbb{R}^n):=\lbrace u \in\mathscr{S}'(\mathbb{R}^n); \;\Lambda^su \in L^2(\mathbb{R}^n) \rbrace, \quad s \in \mathbb{R},\] and endowed with the inner product and the norm given respectively by \[(u,v)_{s}:=(\Lambda^s u, \Lambda^s v)_0, \;\;\|u\|_{s}:=(u,u)^{1/2}_{s}, \quad u,v \in H^s(\mathbb{R}^n).\] Recall also that the space \(H^\infty(\mathbb{R}^n)\) and \(H^{-\infty}(\mathbb{R}^n)\) are defined by \[\label{eq46defHinfty-infty} H^\infty(\mathbb{R}^n):=\bigcap_{s \in \mathbb{R}}H^s(\mathbb{R}^n), \quad H^{-\infty}(\mathbb{R}^n):=\bigcup_{s \in \mathbb{R}} H^s(\mathbb{R}^n),\tag{6}\] respectively. We conclude this subsection by summarizing the following fundamental results of the standard pseudodifferential calculus (cf. Section 2.1 in [5]) that we will use in the next sections.
Theorem 1. Let \(a \in S^m\), with \(m \in \mathbb{R}\), and let \(s \in \mathbb{R}\). Then \(\mathrm{Op}(a)\) extends to a bounded linear operator from \(H^{s+m}(\mathbb{R}^n)\) to \(H^s(\mathbb{R}^n)\) and there exist \(k=k(n,m,s) \in \mathbb{N}_0\) and \(C=C(n,m,s)>0\) such that \[\|\mathrm{Op}(a)u\|_s \leq C |a|_k^{(m)}\|u\|_{s+m}, \quad u \in H^{s+m}(\mathbb{R}^n).\]
Theorem 3. Let \(a \in S^{m_1}\), \(b \in S^{m_2}\), with \(m_1,m_2 \in \mathbb{R}\). Then \[\mathrm{Op}(a)\mathrm{Op}(b)=\mathrm{Op}(ab)+\mathrm{Op}(r_{m_1+m_2-1}),\] where \(r_{m_1+m_2-1} \in S^{m_1+m_2-1}\) and, for each \(k\in \mathbb{N}_0\), there exists \(k_1\in \mathbb{N}_0\) and \(c_1>0\), depending on \(m_1,m_2,k\), such that \[|r_{m_1+m_2-1}|^{(m_1+m_2-1)}_{k}\leq c_1 \, |a|_{k_1}^{(m_1)}|b|_{k_1}^{(m_2)}.\] In particular, \[[\mathrm{Op}(a),\mathrm{Op}(b)]=\mathrm{Op}(i^{-1}\lbrace a, b \rbrace)+\mathrm{Op}(r_{m_1+m_2-2}),\] with \(r_{m_1+m_2-2} \in S^{m_1+m_2-2}\) and, for each \(k\in \mathbb{N}_0\), there exists \(k_2\in \mathbb{N}_0\) and \(c_2>0\), depending once again on \(m_1,m_2,k\), such that \[|r_{m_1+m_2-2}|^{(m_1+m_2-2)}_{k}\leq c_2 \, |a|_{k_2}^{(m_1)}|b|_{k_2}^{(m_2)}.\]
Theorem 2. Let \(a \in S^m\), with \(m \in \mathbb{R}\). Then \[(\mathrm{Op}(a))^\ast=\mathrm{Op}(\bar{a})+\mathrm{Op}(r_{m-1}),\] with \(r_{m-1} \in S^{m-1},\) and for each \(k \in \mathbb{N}_0\) there exists \(k' \in \mathbb{N}_0\) and \(c'>0\), depending on \(m\) and \(k\), such that \[|r_{m-1}|^{(m-1)}_{k}\leq c'|a|^{(m)}_{k'}.\]**
Theorem 4. Let \(a \in S^1\) be such that there exists \(R>0\) so that \[\mathrm{Re}(a)(x,\xi) \geq 0, \quad \text{for} \;\;|\xi|\geq R.\] Then, there exist \(k=k(n) \in \mathbb{N}_0\) and \(C=C(n,R)>0\) such that \[\mathrm{Re} \, (\mathrm{Op}(a)u,u)_0 \geq -C |a|^{(1)}_k\|u\|_0^2, \quad u \in \mathscr{S}(\mathbb{R}^n).\]
In this subsection we establish several useful results concerning the regularisation of both the Cauchy data and the coefficients by means of a mollifier. We also specify the mollifier employed throughout the paper and the scale, together with their basic properties.
Let \(\varphi \in \mathscr{S}(\mathbb{R}^n)\) be such that \(\varphi \geq 0\) and \(\int_{\mathbb{R}^n}\varphi(x)dx=1\). Furthermore, let \(\omega=\omega(\varepsilon)\), with \(\varepsilon \in (0,1]\), be a positive scale, i.e. \(\omega\) is a positive and bounded function, such that \(\omega(\varepsilon) \rightarrow 0\) as \(\varepsilon \rightarrow 0^+\) and \(\omega(\varepsilon)\ge c_k\varepsilon^{k}\) for some \(k\in\mathbb{N}_0\) and \(c_k>0\). It is clearly not restrictive to assume that \(\omega(\varepsilon)<1\). Accordingly, we set \[\label{eq46defphiepsilon} \varphi_{\omega(\varepsilon)}(x):=\frac{1}{(\omega(\varepsilon))^n} \varphi\Bigl(\frac{x}{\omega(\varepsilon)}\Bigr), \quad x \in \mathbb{R}^n, \;\;\varepsilon \in (0,1].\tag{7}\]
In this work we will assume (together with other structural assumptions) that the coefficients of the principal part \(A\) of the operator \(P\) defined in 2 belong to the Sobolev space \[W^{1,\infty}(\mathbb{R}^n):=\lbrace u \in L^\infty(\mathbb{R}^n); \quad \partial_{x_k} u \in L^\infty(\mathbb{R}^n), \;\forall k=1,\dots,n \rbrace,\] endowed with the norm \[\|u\|_{W^{1,\infty}}:=\max_{|\alpha|\leq1}\| \partial^\alpha_{x}u \|_{L^\infty}.\]
The following result will be useful for studying the non-smooth coefficients of our operator 2 .
Proposition 5. Let \(v\in L^\infty(\mathbb{R}^n)\) and \(\varphi_{\omega(\varepsilon)}\) defined as in 7 . Hence, for all \(\beta\in\mathbb{N}_0^n\) there exists \(C_1=C_1(\|v\|_{L^\infty},\beta)>0\) such that \[\label{eq46v} |\partial^\beta_x(v\ast\varphi_{\omega(\varepsilon)})(x)|\le C_1\omega(\varepsilon)^{-|\beta|},\qquad{(1)}\] holds uniformly in \(\varepsilon\in(0,1]\) and \(x\in\mathbb{R}^n\).
In particular, if \(w \in W^{1,\infty}(\mathbb{R}^n)\), we have that for all \(\beta \in \mathbb{N}_0^n\) there exists \(C_2=C_2(\beta,\|w\|_{W^{1,\infty}})>0\) such that \[\label{eq46w} |\partial^\beta_x(w\ast\varphi_{\omega(\varepsilon)})(x)|\le C_2\omega(\varepsilon)^{-|\beta|+1},\qquad{(2)}\] uniformly in \(\varepsilon\in(0,1]\) and \(x\in\mathbb{R}^n\).
Proof. To prove ?? it is sufficient to note that \[\partial_x^\beta (v \ast \varphi_{\omega(\varepsilon)})(x)= (v \ast \partial_y^{\beta} \varphi_{\omega(\varepsilon)})(x), \quad x \in \mathbb{R}^n,\] and then, by Young’s convolution inequality, \[\|\partial^\beta_x(v \ast \varphi_{\omega(\varepsilon)})\|_{L^\infty} \leq \omega(\varepsilon)^{-|\beta|}\|v\|_{L^\infty}\|\partial_y^\beta \varphi\|_{L^1}, \quad \forall \varepsilon\in (0,1].\] Moreover, if \(\beta \in \mathbb{N}_0^n\), \(\beta=\beta'+\beta''\), \(|\beta'|=1\), \(|\beta''|=|\beta|-1\), we have \[\partial_x^\beta (w \ast \varphi_{\omega(\varepsilon)})(x)= (\partial_y^{\beta'}w \ast \partial_y^{\beta''} \varphi_{\omega(\varepsilon)})(x), \quad x \in \mathbb{R}^n.\] Hence ?? follows by \[\|\partial^\beta_x(w \ast \varphi_{\omega(\varepsilon)})\|_{L^\infty} \leq \omega(\varepsilon)^{-|\beta|+1}\|w\|_{W^{1,\infty}}\|\partial_y^{\beta''}\varphi\|_{L^1}, \quad \forall \varepsilon\in (0,1].\] ◻
Furthermore, the following regularisation argument will play a key role in establishing consistency with the classical theory.
Proposition 6. Let \(v \in H^s(\mathbb{R}^n)\), with \(s \in \mathbb{R}\), and let \(\varphi_{\omega(\varepsilon)}\) be as in 7 . Then, for all \(\ell \in \mathbb{N}_0\) there exists a constant \(C=C(\ell,\varphi)>0\) such that the estimate \[\|v \ast \varphi_{\omega(\varepsilon)}\|_{s+\ell} \leq C \omega(\varepsilon)^{-\ell}\|v\|_{s},\] holds uniformly in \(\varepsilon\).
Proof. As before, we denote by \(C\) a positive constant, possibly changing from line to line, independent of \(\varepsilon\). Since \(\Lambda^{s+\ell}=\mathscr{F}_{\xi \rightarrow x}^{-1} \langle \xi \rangle^{s+\ell} \mathscr{F}_{y \rightarrow \xi}\), by Plancherel’s theorem, we have \[\|v \ast \varphi_{\omega(\varepsilon)}\|_{s+\ell}^2 \leq C \|\langle \xi \rangle^{s+\ell} \widehat{v \ast \varphi_{\omega(\varepsilon)}} \|_0^2 \leq C \|(\langle \xi \rangle^\ell \widehat{\varphi_{\omega(\varepsilon)}})(\langle \xi \rangle^s \hat{v})\|_0^2,\] and then, by Hölder’s inequality (applied with \(p=1\) and \(q=+\infty\)), \[\|v \ast \varphi_{\omega(\varepsilon)} \|_{s+\ell}^2\leq C \|\langle \xi \rangle^{2\ell}\widehat {\varphi_{\omega(\varepsilon)}}^2\|_{L^\infty} \|v\|_s^2.\] Therefore, since \(\varphi \in \mathscr{S}(\mathbb{R}^n)\) and \(\widehat{\varphi_{\omega(\varepsilon)}}(\xi)=\hat{\varphi}(\omega(\varepsilon) \xi)\) we obtain \[\|v \ast \varphi_{\omega(\varepsilon)} \|_{s+\ell}\leq C \omega(\varepsilon)^{-\ell} \|v\|_s.\] ◻
We recall here the notions of \(H^s\)- and \(H^\infty\)-moderate nets of functions (cf. [7], [15],[14]) that, in the next sections, will lead to the notion of \(H^s\)- and \(H^\infty\)-very weak solution of the problem 1 . Furthermore, we also recall the notion of negligible net, that will be useful to treat uniqueness in the context of very weak solutions. In what follows, we consider \(T>0\) and \(s \in \mathbb{R}\) as fixed.
Definition 7. Let \((v_\varepsilon)_\varepsilon \in \lbrace C([0,T];H^s(\mathbb{R}^n) \rbrace^{(0,1]}\), with \(s \in \mathbb{R}\). We say that the net \((v_\varepsilon)_\varepsilon\) is \(H^s\)-moderate if there exist \(N \in \mathbb{N}_0\) and \(C>0\) such that \[\|v_\varepsilon(t,\cdot) \|_{H^s} \leq C \varepsilon^{-N}, \quad \forall t \in [0,T], \;\;\forall \varepsilon \in (0,1].\] Moreover, if \((v_\varepsilon)_\varepsilon \in \lbrace C([0,T];H^\infty(\mathbb{R}^n) \rbrace^{(0,1]}\) we say that \((v_\varepsilon)_\varepsilon\) is \(H^\infty\)-moderate if for each \(s \in \mathbb{R}\) there exist \(N \in \mathbb{N}_0\) and \(C>0\) such that \[\|v_\varepsilon(t,\cdot) \|_{H^s} \leq C \varepsilon^{-N}, \quad \forall t \in [0,T], \;\;\forall \varepsilon \in (0,1].\]
This notion leads to the notion of \(H^s\)-moderate and \(H^\infty\)-moderate regularisations of a distribution.
Definition 8. Let \(v \in C([0,T];\mathscr{D}'( \mathbb{R}^n))\). Moreover, for \(\varepsilon \in (0,1]\) and \(t \in (0,1]\), define \(v_\varepsilon(t,x)=(v(t,\cdot) \ast \varphi_{\varepsilon})(x)\), with \(\varphi \in \mathscr{S}(\mathbb{R}^n)\) satisfying \(\int_{\mathbb{R}^n}\varphi(x)dx=1\), and \(\varphi_\varepsilon=\varepsilon^{-n}\varphi(\cdot/\varepsilon)\). If \((v_\varepsilon)_\varepsilon\) is \(H^s\)-moderate (resp. \(H^\infty\)-moderate), we say that \((v_\varepsilon)_\varepsilon\) is an \(H^s\)-moderate regularisation (resp. \(H^\infty\)-moderate regularisation) of \(v\).
Remark 9. Note that, if \(v \in H^{-\infty}(\mathbb{R}^n)\) (recall 6 ) by repeating essentially the same proof of Proposition 6, we get that the net \((v_\varepsilon)_\varepsilon\), defined by \(v_\varepsilon=v \ast \varphi_\varepsilon\), is an \(H^\infty\)- regularisation of \(v\).
We conclude this section with the definition of negligible nets.
Definition 10. Let \((v_\varepsilon)_\varepsilon \in \lbrace C([0,T];H^s(\mathbb{R}^n) \rbrace^{(0,1]}\), with \(s \in \mathbb{R}\). We say that \((v_\varepsilon)_\varepsilon\) is \(H^s\)-negligible if for any \(q \in \mathbb{N}_0\) there exists \(C>0\) such that \[\|v_\varepsilon(t,\cdot) \|_{H^s} \leq C \varepsilon^{q}, \quad \forall t \in [0,T], \;\;\forall \varepsilon \in (0,1].\] Moreover, if \((v_\varepsilon)_\varepsilon \in \lbrace C([0,T];H^{\infty}(\mathbb{R}^n) \rbrace^{(0,1]}\), we say that the net \((v_\varepsilon)_\varepsilon\) is \(H^\infty\)-negligible if for any \(s \in \mathbb{R}\) and any \(q \in \mathbb{N}_0\) there exists \(C>0\) such that \[\|v_\varepsilon(t,\cdot) \|_{H^s} \leq C \varepsilon^{q}, \quad \forall t \in [0,T], \;\;\forall \varepsilon \in (0,1].\]
startsection section1@-3.5ex plus -1ex minus -.2ex2.3ex plus .2exUltrahyperbolic operators with singular coefficients The goal of this section is to study the main properties of the singular version of ultrahyperbolic operators introduced in [5] (cf. [4]), namely the higher-order part of the operator \(P\) defined in 2 . We first introduce the net of regularised operators associated with such a singular operator and state the main assumptions. We then prove a new version of Doi’s lemma (see Lemma 17 below) in our setting, which relies on the construction of a suitable net of functions \((q_\varepsilon)_\varepsilon\), provided by Proposition 16.
In this subsection we consider a class of operators with non-smooth coefficients that can be formally written as \[\label{eq46A} A=\sum_{i,j=1}^n D_{x_i}(a_{ij}(x)D_{x_j}),\tag{8}\] where the coefficients \(a_{ij}\) are given by \[\label{eq46coeffA} a_{ij}(x)=c_{ij}+\tilde{a}_{ij}(x), \quad i,j=1,\dots,n, \quad x \in \mathbb{R}^n,\tag{9}\] with \(c_{ij} \in \mathbb{R}\) and \(\tilde{a}_{ij} \in W^{1,\infty}(\mathbb{R}^n)\) real valued, for all \(i,j=1,\dots,n\), and satisfying suitable conditions. In what follows, we also denote by \(\mathcal{A}(x)=(a_{ij}(x))_{i,j=1,\dots,n}\) the coefficients matrix.
Remark 11. We recall that the space \(W^{1,\infty}(\mathbb{R}^n)\) essentially coincides with the space of bounded Lipschitz functions. Therefore, one of the interests of our theory is that it allows us to establish well-posedness results for Cauchy problems induced by variable-coefficient operators whose principal part is obtained as a bounded Lipschitz perturbation of a constant-coefficient operator and, as we will see below, with much more singular lower-order terms.
Our purpose is now to construct a net of operator \((A_\varepsilon)_\varepsilon\) associated with \(A\) obtained via a regularisation argument. In detail, we will comvolve with a mollifier \(\varphi_{\omega(\varepsilon)}\) defined (as in 7 ) by \[\varphi_{\omega(\varepsilon)}(x):=\frac{1}{(\omega(\varepsilon))^n} \varphi\Bigl(\frac{x}{\omega(\varepsilon)}\Bigr), \quad x \in \mathbb{R}^n, \;\;\varepsilon \in (0,1],\] where, recall \(\varphi \in \mathscr{S}(\mathbb{R}^n)\), \(\varphi \geq 0\), \(\int\varphi=1\) and \(\omega\) is a positive scale, as specified in Subsection 0.2.
For all \(i,j=1,\dots,n\), we define the the \(\varepsilon\)-regularised coefficients of \(A\), defined in 8 , as \[\label{eq46regcoeff} a_{ij,\varepsilon}(x):=(a_{ij} \ast \varphi_{\omega(\varepsilon)})(x), \quad x \in \mathbb{R}^n, \;\;i,j=1,\dots,n, \;\;\varepsilon \in (0,1].\tag{10}\] We thus define, for \(\varepsilon \in (0,1]\), the \(\varepsilon\)-regularised operator \[\label{eq46Aepsilon} A_\varepsilon:=\sum_{i,j=1}^nD_{x_i}(a_{ij,\varepsilon}(x)D_{x_j}),\tag{11}\] and denote by \(\mathcal{A}_\varepsilon(x)=(a_{ij,\varepsilon}(x))_{i,j=1}^n\) the matrix of \(\varepsilon\)-regularised coefficients. Finally, \((A_\varepsilon)_\varepsilon\) is the net of regularised operators associated with \(A\).
Proposition 12. Let \(A_\varepsilon\) be the operator defined in 11 , with \(\varepsilon\in (0,1]\). Then, \(A_\varepsilon=\mathrm{Op}(a_\varepsilon)\) is an operator with symbol \(a_\varepsilon(x,\xi)\) fulfilling the following statement: for all \(\alpha,\beta \in \mathbb{N}_0^n\) there exists a constant \(C_{\alpha,\beta}>0\) such that \[\label{eq46conda} |\partial_x^\beta\partial_\xi^\alpha a_{\varepsilon}(x,\xi)|\leq C_{\alpha,\beta} \, \omega(\varepsilon)^{-|\beta|}\langle \xi \rangle^{2-|\alpha|}, \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n.\qquad{(3)}\]
Proof. Note that \(A_\varepsilon=\mathrm{Op}(a_\varepsilon)\), where \[\label{eq46defaeps} \begin{align} a_\varepsilon(x,\xi)&=\sum_{i,j=1}^n a_{ij,\varepsilon}(x)\xi_i\xi_j+\sum_{i,j=1}^n (D_{x_i}a_{ij,\varepsilon})(x)\xi_j, \\ &=:a_{2,\varepsilon}(x,\xi)+a_{1,\varepsilon}(x,\xi), \quad (x,\xi)\in \mathbb{R}^{n}\times \mathbb{R}^n, \end{align}\tag{12}\] and \(a_{ij,\varepsilon}\) defined as in 10 . Then, since \(\tilde{a}_{ij} \in W^{1,\infty}(\mathbb{R}^n)\) for all \(i,j=1,\dots,n\), by ?? we obtain that for all \(\alpha,\beta \in \mathbb{N}_0^n\) there exists \(C_{\alpha,\beta}'>0\) such that
\[\label{eq46conda2} |\partial_x^\beta\partial_\xi^\alpha a_{2,\varepsilon}(x,\xi)|\leq C_{\alpha,\beta}' \, \omega(\varepsilon)^{-|\beta|+1}\langle \xi \rangle^{2-|\alpha|}, \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n,\tag{13}\] and, similarly (since \(D_{x_i}a_{ij} \in L^\infty(\mathbb{R}^n)\) for all \(i,j\)) by ?? , there exists \(C_{\alpha,\beta}''>0\) such that \[\label{eq46conda1} |\partial_x^\beta\partial_\xi^\alpha a_{1,\varepsilon}(x,\xi)|\leq C_{\alpha,\beta}'' \, \omega(\varepsilon)^{-|\beta|}\langle \xi \rangle^{1-|\alpha|}, \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n.\tag{14}\] Therefore, ?? easily follows from 12 . ◻
In order to study the well-posedness (see Subsection 0.6 below) of the Cauchy problem 1 associated with the operator \(P\), we assume the hypotheses below on the net of regularised operators \((A_\varepsilon)_\varepsilon\), representing the regularised principal part of \(P\). These conditions, together with further assumptions on the regularised lower-order parts of \(P\), are satisfied for a class of (singular) operators that motivated our work (see Theorem 20). We assume the following hypotheses.
1. For each \(\varepsilon \in (0,1]\) the coefficients matrix \(\mathcal{A}_\varepsilon\) is real and symmetric.
2. There exists a universal constant \(\mu>0\) such that for any \(\varepsilon \in (0,1]\) one has \[\mu^{-1}|\xi| \leq |\mathcal{A}_\varepsilon(x) \cdot \xi| \leq \mu |\xi|, \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n.\]
3. There exist \(\nu>0\) sufficiently small and \(N \in \mathbb{N}\), \(N>1\), such that, for any \(\varepsilon \in (0,1]\) one has \[|\partial_{x_k} a_{ij,\varepsilon}(x)| \leq \nu \langle x \rangle^{-N}, \quad \forall x \in \mathbb{R}^n, \;\;i,j,k=1,\dots,n.\]
As anticipated above, let us emphasise that, once these conditions are required on the net \((A_\varepsilon)_\varepsilon\), the operator \(A\) can be interpreted as the "singular-coefficients" version of the ultrahyperbolic operators studied in [4] and [5]. We conclude this subsection with a few comments and motivations related to the hypotheses (H1)-(H3).
Remark 13. In the first place, it is important to note that the ultrahyperbolic condition 2 (as named in [5]) is actually a non-degenerate condition. In particular, let us consider \[\label{eq46aquad} a(x,\xi)=\langle \mathcal{A}(x)\cdot \xi,\xi \rangle, \quad (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n,\qquad{(4)}\] where \(\mathcal{A}(x)=(a_{ij}(x))\) and \(a_{ij} \in C^\infty(\mathbb{R}^n)\) real valued with bounded derivatives at any order, for all \(i,j=1,\dots,n\).
If the symbol \(a(x,\xi)\) satisfies the ellipticity condition \[\label{eq46ellsymbol} C^{-1} |\xi|^2 \leq a(x,\xi) \leq C |\xi|^2, \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n,\qquad{(5)}\] for some \(C>0\), by Cauchy-Schwartz inequality (recall ?? ), we get \[C^{-1}|\xi|^2 \leq a(x,\xi) \leq |\mathcal{A}(x)\cdot \xi | |\xi| , \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n,\] which implies \[|\mathcal{A}(x)\cdot \xi | \geq C^{-1}|\xi|, \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n.\] Moreover, by the bounded assumption on the coefficients \(a_{ij}(x)\), possibly by enlarging the constant \(C>0\), we also have \[|\mathcal{A}(x)\cdot \xi | \leq C|\xi|, \quad (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n.\] Therefore, the variable coefficient elliptic operators, whose symbol satisfies ?? , fulfills the "ultrahyperbolic condition" \[C^{-1}|\xi|\leq |\mathcal{A}(x)\cdot \xi | \leq C|\xi|, \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n.\] This means that our theory includes the non-smooth variable coefficients Schrödinger operator of the form \(P=D_t-A(x,D_x)+\text{l.o.t.}\) (where l.o.t. means lower order terms), with \(A\) non-smooth elliptic operator.
Remark 14. As for the hypothesis 3, it is immediate to construct an operator having coefficients for which the net of regularised coefficients satisfy 3. Explicitly, if we consider \((b_{ij})_{i,j=1}^n\) where \(b_{ij} \in W^{1,\infty}(\mathbb{R}^n)\) such that \(\|b_{ij}\|_{W^{1,\infty}}\leq \nu\), for \(\nu>0\) sufficiently small, setting \[\tilde{a}_{ij}(x):=\langle x \rangle^{-N}b_{ij}(x), \quad x \in \mathbb{R}^n, \;\;i,j=1,\dots,n,\] we get that the coefficients \(a_{ij,\varepsilon}\) defined as in 10 satisfy 3.
Indeed, using the Peetre’s inequality \(\langle x-y \rangle^{-N} \leq C_N \langle x \rangle^{-N}\langle y \rangle^{N}\), for all \(\varepsilon \in (0,1]\) and for all \(i,j,k=1,\dots,n\), we have \[\label{eq46peetre} \begin{align} |\partial_{x_k}\tilde{a}_{ij,\varepsilon}(x)|&=\Bigl|\int_{\mathbb{R}^n}\partial_{x_k}(\langle x-y \rangle^{-N}b_{ij}(x-y))\varphi_{\omega(\varepsilon)}(y)dy\Bigr|\\ &\leq \int_{\mathbb{R}^n}|\partial_{x_k}(\langle x-y \rangle^{-N}b_{ij}(x-y))\varphi_{\omega(\varepsilon)}(y)|dy \\ &\leq C_{N}\langle x \rangle^{-N}\|b_{ij}\|_{W^{1,\infty}}\int_{\mathbb{R}^n}\langle y \rangle^{N}\varphi_{\omega(\varepsilon)}(y)dy \\ &\leq C_{N,\varphi} \, \nu \langle x \rangle^{-N}, \quad \forall x \in \mathbb{R}^n, \end{align}\qquad{(6)}\] for some \(C_{N,\varphi}>0\), uniformly in \(\varepsilon\in (0,1]\).
We finally present some examples of operators that fall under the scope of our study. In the next example, we consider models that generalise non-degenerate smooth operators, in the singular setting, defined through a diagonal coefficients matrix (see ?? ). In this context, depending on the value of the constant \(c_2\) below, we recover an elliptic operator with non-smooth coefficients (if \(c_2 > 0\)) and a "hyperbolic" operator (if \(c_2 < 0\)).
Example 15. Let us consider on \(\mathbb{R}^2\) the operator \[A=\sum_{i,j=1}^2D_{x_i}(a_{ij}(x)D_{x_j}),\] defined by the matrix \[\label{eq46diagmatrix} \mathcal{A}(x)= \begin{pmatrix} a_{11}(x) & 0 \\ 0 & a_{22}(x) \end{pmatrix}, \quad x \in \mathbb{R}^2,\qquad{(7)}\] where \[\begin{align} a_{11}(x)&=c_1+\tilde{a}_{11}(x),\\ a_{22}(x)&=c_2+\tilde{a}_{22}(x), \end{align}\] with \(c_1>0\), \(c_2\in \mathbb{R}\setminus \lbrace 0 \rbrace\), and \(\tilde{a}_{11}(x), \tilde{a}_{22}(x) \in W^{1,\infty}(\mathbb{R}^2)\) real functions, such that \[\begin{align} & 0\leq |\tilde{a}_{jj}(x)| \leq \frac{|c_j|}{2}, \quad \forall x \in \mathbb{R}^2, \;\;j=1,2, \\ & |\partial_{x_i}\tilde{a}_{jj}(x)| \leq \nu \langle x \rangle^{-N}, \quad \text{for a.e.} \; x \in \mathbb{R}^2, \;\;i,j=1,2, \end{align}\] for some \(N>1\) and some \(\nu>0\) sufficiently small.
Under these assumptions, we get that 1 is trivially satisfied. As for 2, for \(\varepsilon \in (0,1]\), we get \[|\mathcal{A}_\varepsilon(x) \cdot \xi |^2=a_{11,\varepsilon}(x)^2\xi_1^2+a_{22,\varepsilon}(x)^2\xi_2^2.\] Hence, for all \((x,\xi) \in \mathbb{R}^n\times \mathbb{R}^n\), \[\begin{align} &|\mathcal{A}_\varepsilon(x) \cdot \xi| \leq \max{\Bigl\lbrace\frac{3}{2}c_1,\frac{3}{2}|c_2|\Bigr\rbrace}|\xi| \\ & |\mathcal{A}_\varepsilon(x) \cdot \xi| \geq \min{\Bigl\lbrace\frac{1}{2}c_1,\frac{1}{2}|c_2|\Bigr\rbrace}|\xi|, \end{align}\] and 2 easily follows. Finally, regarding 3, by reasoning as in ?? we have \[\label{eq46smallcoeffex} \begin{align} |\partial_{x_i}a_{jj,\varepsilon}(x)| &\leq \int_{\mathbb{R}^2}|\partial_{x_i}a_{jj}(x-y)||\varphi_{\omega(\varepsilon)}(y)|dy \\ & \leq \nu \int_{\mathbb{R}^2}\langle x-y \rangle^{-N}|\varphi_{\omega(\varepsilon)}(y)|dy \\ & \leq C_N \, \nu \int_{\mathbb{R}^2} \langle x \rangle^{-N}\langle y \rangle^N |\varphi_{\omega(\varepsilon)}(y)|dy \\ & \leq C_{N,\varphi} \, \nu \langle x \rangle^{-N}, \quad \forall x \in \mathbb{R}^2, \;\;i,j=1,2, \end{align}\qquad{(8)}\] for some \(C_{N,\varphi}>0\), uniformly in \(\varepsilon\in (0,1]\).
Next, for the operators under consideration the following fundamental Proposition holds (cf. [4]).
Proposition 16. Let \(A\) be an operator as in 8 , with coefficients of the form 9 and let \((A_\varepsilon)_\varepsilon\) be the corresponding net of regularised operators. If 1, 2 and 3 hold, then there exists a net of real-valued functions \((q_\varepsilon)_\varepsilon\), with \(q_\varepsilon \in C^\infty(\mathbb{R}^n \times \mathbb{R}^n)\) that satisfies the following:
(i) For each \(\alpha,\beta \in \mathbb{N}_0^n\) there exists a constant \(C_{\alpha,\beta}>0\) such that \[\label{prop46q} | \partial_\xi^\alpha\partial_x^\beta q_\varepsilon(x,\xi)|\leq \left\{ \begin{array}{ll} C_{\alpha,\beta} \langle x\rangle \langle \xi \rangle^{-|\alpha|}, & \text{if}\,\,|\beta|=0, \\ C_{\alpha,\beta} \omega(\varepsilon)^{-|\beta|+1} \langle \xi \rangle^{-|\alpha|}, & \text{if}\,\,|\beta|\geq 1 , \end{array}\right.\qquad{(9)}\] for all \((x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n\).
(ii) There exist \(C_1, C_2>0\) such that \[\begin{align} H_{a_{2,\varepsilon}}q_\varepsilon(x,\xi)\geq C_1|\xi|-C_2, \end{align}\] for all \((x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n\), where \(H_{a_2,\varepsilon}\) is the Hamilton vector field defined in 4 .
Proof. For each \(\varepsilon \in (0,1]\) let us define (with \(\mu>0\) as in 2 and \(C_1>0\)) \[q_\varepsilon(x,\xi):=C_1\mu^2\langle \xi \rangle^{-1}\sum_{j=1}^n x_j\partial_{\xi_j}a_{2,\varepsilon}(x,\xi), \quad (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n,\] where recall \(a_{2,\varepsilon}\) is defined in 12 . With this definition of \(q_\varepsilon\), condition \((i)\) is satisfied because of the properties of \(a_{2,\varepsilon}\) (see 3 and 13 ).
Moreover, using the expression 5 , we have \[\begin{align} H_{a_2,\varepsilon}q_\varepsilon(x,\xi)&= (\langle \nabla_\xi a_{2,\varepsilon},\nabla_x q_\varepsilon \rangle - \langle \nabla_x a_{2,\varepsilon},\nabla_\xi q_\varepsilon \rangle)(x,\xi) \\ &=C_1\mu^2|\nabla_\xi a_{2,\varepsilon}(x,\xi)|^2\langle \xi \rangle ^{-1}+C_1\mu^2\langle \xi \rangle ^{-1}\sum_{j,k=1}^n x_j\partial_{\xi_k}a_{2,\varepsilon}(x,\xi)\partial_{\xi_j}\partial_{x_k}a_{2,\varepsilon}(x,\xi) \, + \\ &\quad \quad -\langle\nabla_x a_{2,\varepsilon}(x,\xi),\nabla_\xi q_{\varepsilon}(x,\xi) \rangle\\ &\geq C_1\mu^2|\nabla_\xi a_{2,\varepsilon}(x,\xi)|^{2}\langle \xi \rangle ^{-1} - C_1\mu^2\langle \xi \rangle ^{-1}\sum_{j,k=1}^n |x_j\partial_{\xi_k}a_{2,\varepsilon}(x,\xi)\partial_{\xi_j}\partial_{x_k}a_{2,\varepsilon}(x,\xi)| \, + \\ &\quad\quad - |\nabla_x a_{2,\varepsilon}(x,\xi)||\nabla_\xi q_{\varepsilon}(x,\xi)|, \quad \forall (x,\xi) \in \mathbb{R}^{2n}. \end{align}\] Hence, since \[|\nabla_\xi a_{2,\varepsilon}(x,\xi)|=2|\mathcal{A}_\varepsilon(x)\cdot \xi|, \quad (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n,\] by 2 and 3 we obtain \[H_{a_{2,\varepsilon}} q_\varepsilon(x,\xi)=\geq 2C_1|\xi|^{2}\langle \xi \rangle ^{-1} -\nu C'\langle \xi \rangle\geq (2C_1-\nu C'')|\xi|-C_2, \quad \forall (x,\xi)\in \mathbb{R}^n\times \mathbb{R}^n,\] for some constants \(C',C'',C_2\) depending on \(\mu,C_1\) and on the (universal) bounds of the coefficients \(a_{ij}\), but not on \(\varepsilon\). Therefore, for \(\nu>0\) sufficiently small \[H_{a_{2,\varepsilon}} q_\varepsilon(x,\xi) \geq C_1 |\xi|-C_2\quad \forall(x,\xi)\in\mathbb{R}^{n}\times \mathbb{R}^n,\] and then \((ii)\). This concludes the proof of the lemma. ◻
We now prove the following generalised version of Doi’s Lemma (cf. [20], [4]).
Lemma 17. Let \(A\) be an operator as in 8 with coefficients of the form 9 , and let \((A_\varepsilon)_\varepsilon\) be the corresponding net of regularised operators. Moreover, assume that there exists a net of real-valued functions \((q_\varepsilon)_\varepsilon\), with \(q_\varepsilon \in C^\infty(\mathbb{R}^n \times \mathbb{R}^n)\) that satisfies the following statements:
(i) For each \(\alpha,\beta \in \mathbb{N}_0^n\) there exists a constant \(C_{\alpha,\beta}>0\) such that \[\label{eq46qip} | \partial_\xi^\alpha\partial_x^\beta q_\varepsilon(x,\xi)|\leq \left\{ \begin{array}{ll} C_{\alpha,\beta} \langle x\rangle \langle \xi \rangle^{-|\alpha|}, & \text{if}\,\,|\beta|=0, \\ C_{\alpha,\beta} \omega(\varepsilon)^{-|\beta|+1} \langle \xi \rangle^{-|\alpha|}, & \text{if}\,\,|\beta|\geq 1 , \end{array}\right.\qquad{(10)}\] for all \((x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n\).
(ii) There exist \(C_1, C_2>0\) such that \[\label{eq46qiip} H_{a_{2,\varepsilon}}q_\varepsilon(x,\xi)\geq C_1|\xi|-C_2,\qquad{(11)}\] for all \((x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n\).
Then, there exists a net of symbols \((d_\varepsilon)_\varepsilon\) such that for all \(\alpha,\beta \in \mathbb{N}_0^n\) \[\label{eq46pSM} |\partial_x^\beta\partial_\xi^\alpha d_\varepsilon(x,\xi)| \leq C_{\alpha,\beta}\omega(\varepsilon)^{-|\beta|+1}\langle \xi \rangle^{-|\alpha|}, \quad \forall (x,\xi) \in \mathbb{R}^n\times \mathbb{R}^n,\, \forall \varepsilon \in (0,1]\qquad{(12)}\] for some \(C_{\alpha,\beta}>0\) and \[\label{eq46pDoi} H_{a_{2,\varepsilon}}d_\varepsilon(x,\xi)\geq \langle x \rangle^{-N}|\xi|-C,\quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n,\, \forall \varepsilon \in (0,1],\qquad{(13)}\] for some \(C>0\) independent of \(\varepsilon\) and \(N\in \mathbb{N}\), \(N>1\).
Proof. In the first place, we note that, by hypothesis ?? , there exists a fixed constant \(K>0\) such that, for each \(\varepsilon \in (0,1]\) one has \[\label{eq46estqepsilonK} |q_\varepsilon(x,\xi)| \leq K \langle x \rangle, \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n.\tag{15}\] Let us define \(\lambda(t) := \langle t \rangle^{-N}\) for \(t \geq 0\), and extend it to \(t < 0\) by setting \(\lambda(t) := \lambda(0)\) if \(t<0\). By Lemma 3.1 in [20] there exists a nonnegative real-valued function \(f \in C^\infty([0,+\infty))\) such that (cf. [4]) \[\label{eq46fdoi1} f'(t) \geq \lambda(K^{-1}t - 10), \quad \forall t \geq 0,\tag{16}\] and \[\label{eq46foi2} |f^{(m)}(t)| \leq C_m \left( \lambda(0) + \int_0^t \lambda(s)ds \right) (1 + t)^{-m}, \quad \forall t \geq 0,\tag{17}\] for some constants \(C_m > 0\). Therefore, by 15 and 16 we have \[\label{eq46fder1} f'(|q_\varepsilon(x,\xi)|) \geq \lambda(K^{-1}|q_\varepsilon(x,\xi)|-10) \geq \lambda (\langle x \rangle -10) \geq \lambda(|x|)=\langle x \rangle^{-N}, \quad \forall (x,\xi) \in \mathbb{R}^n\times \mathbb{R}^n.\tag{18}\] Now, we take \(\delta>0\) sufficiently small to be fixed later, and \(\phi \in C^\infty(\mathbb{R})\) such that \(\phi(t)=0\), if \(t\leq 1\), \(\phi(t)=1\) if \(t \geq 2\), and \(\phi'(t) \geq 0\) on \(\mathbb{R}\). We set \(\phi_+(t):=\phi(t/\delta)\), \(\phi_-(t):=\phi(-t/\delta)\) and \(\phi_0:=1-\phi_+-\phi_-\). For \(\varepsilon\in (0,1]\) we define \(\psi_{+,\varepsilon},\psi_{-,\varepsilon},\psi_{0,\varepsilon} \in S^0\) as \[\psi_{0,\varepsilon}(x,\xi):=\phi_0(q_\varepsilon(x,\xi)/\langle x \rangle), \quad \psi_{+,\varepsilon}(x,\xi):=\phi_+(q_\varepsilon(x,\xi)/\langle x \rangle), \quad \psi_{-,\varepsilon}(x,\xi):=\phi_{-}(q_\varepsilon(x,\xi)/\langle x \rangle).\] By 16 , for all \(\alpha,\beta \in \mathbb{N}_0^n\) there exists a constant \(C_{\alpha,\beta}>0\) such that \[|\partial_x^\beta \partial_\xi^{\alpha}f(|q_\varepsilon(x,\xi)|)|\leq C_{\alpha,\beta}\omega(\varepsilon)^{-|\beta|+1}\langle \xi \rangle^{-|\alpha|} \quad \text{on} \,\,\, \mathrm{supp} \, \psi_{+,\varepsilon} \cup \mathrm{supp} \, \psi_{-,\varepsilon}.\]
Next, for \(\varepsilon \in (0,1]\), we define \(d_\varepsilon \in S^0\) as \[\label{Hp} d_\varepsilon(x,\xi):=\frac{q_\varepsilon(x,\xi)}{\langle x \rangle}\psi_{0,\varepsilon}(x,\xi)+(f(|q_\varepsilon(x,\xi)|)+2\delta)(\psi_{+,\varepsilon}(x,\xi)-\psi_{-,\varepsilon}(x,\xi)), \quad (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n.\tag{19}\] By this definition of \(d_\varepsilon\) we get that ?? is satisfied due to the property of \(q_\varepsilon\). Hence, the goal is to prove ?? .
In the first place, we note that by hypotheses (i) and (ii) on \((q_\varepsilon)_\varepsilon\), by choosing \(\delta>0\) sufficiently small, for all \(\varepsilon \in (0,1]\) we have \[H_{a_{2,\varepsilon}}\Bigl(\frac{q_\varepsilon}{\langle x \rangle}\Bigr)=\frac{H_{a_{2,\varepsilon}}q_\varepsilon}{\langle x \rangle}-\frac{q_\varepsilon}{\langle x \rangle}\frac{x \cdot \nabla_\xi a_{2,\varepsilon}}{\langle x \rangle^2}\geq C_1' \frac{|\xi|}{\langle x \rangle}-C_2' \quad \text{on}\,\, \mathrm{supp} \, \psi_{0,\varepsilon},\] for some universal constants \(C_1',C_2'>0\), independent of \(\varepsilon\). Hence, by using the properties of \(f\) and the inclusions \(\mathrm{supp} \, \phi'_+, \mathrm{supp} \, \phi'_- \subseteq \mathrm{supp}\,\phi_0\), for all \(\varepsilon \in (0,1]\) we get \[\begin{align} H_{a_{2,\varepsilon}}d_\varepsilon&=H_{a_{2,\varepsilon}}\Bigl(\frac{q_\varepsilon}{\langle x \rangle}\Bigr)\psi_{0,\varepsilon}+ f'(\lvert q_\varepsilon\rvert)(H_{a_{2,\varepsilon}}q_\varepsilon)(\psi_{+,\varepsilon} +\psi_{-,\varepsilon})\\ &\quad +\Bigl(f(\lvert q_\varepsilon\rvert)+2\delta-\frac{\lvert q_\varepsilon\rvert}{\langle x \rangle}\Bigr)\Bigl(\phi_+'\Bigl(\frac{q_\varepsilon}{\langle x \rangle}\Bigr)-\phi_-'\Bigl(\frac{q_\varepsilon }{\langle x \rangle}\Bigr)\Bigr)H_{a_{2,\varepsilon}}\Bigl(\frac{q_\varepsilon}{\langle x \rangle}\Bigr) \nonumber\\ &\geq H_{a_{2,\varepsilon}}\Bigl(\frac{q_\varepsilon}{\langle x \rangle}\Bigr)\psi_{0,\varepsilon}+ f'(\lvert q_\varepsilon\rvert)(H_{a_{2,\varepsilon}}q_\varepsilon)(\psi_{+,\varepsilon} +\psi_{-,\varepsilon}) -C_3,\label{eq46Hap} \end{align}\tag{20}\] for some \(C_3>0\).
To estimate the second term on the RHS of 20 we use hypothesis (ii) and 18 . Putting everything together we obtain \[\begin{align} H_{a_{2,\varepsilon}}d_\varepsilon(x,\xi)&\geq C_4\Big(\langle x \rangle^{-1} \psi_{0,\varepsilon}+\langle x\rangle^{-N}(\psi_{+,\varepsilon}+\psi_{-,\varepsilon})\Big)\lvert\xi\rvert-C_5, \\ & \geq C_4 \langle x \rangle^{-N}(\psi_{0,\varepsilon}+\psi_{+,\varepsilon}+\psi_{-,\varepsilon})\lvert\xi\rvert-C_5\\ & =C_4\langle x \rangle^{-N}|\xi|-C_5, \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n, \end{align}\] for some new constants \(C_4,C_5>0\). In conclusion, possibly rescaling the symbol \(d_\varepsilon\), we showed that there exists a constant \(C>0\) such that for all \(\varepsilon \in (0,1]\) \[H_{a_{2,\varepsilon}}d_\varepsilon\geq \langle x \rangle^{-N}\lvert\xi\rvert-C, \quad \forall (x, \xi) \in \mathbb{R}^n\times \mathbb{R}^n.\] ◻
Remark 18. Let us point out that, as a consequence of Proposition 16, we obtain that, if 1, 2, 3 hold, there exists a net \((d_\varepsilon)_\varepsilon\)of symbols of order \(0\) such that ?? and ?? are satisfied.
Remark 19. Note also that, possibly by rescaling the symbol \(d_\varepsilon\) with a universal constant (the same for each \(\varepsilon \in (0,1]\)), instead of ?? we obtain \[\label{eq46pDoi2} H_{a_{2,\varepsilon}}d_\varepsilon\geq C_1 \langle x \rangle^{-N} |\xi| - C', \quad \forall (x, \xi) \in \mathbb{R}^n \times \mathbb{R}^n,\qquad{(14)}\] where \(C_1, C' > 0\) are universal constants (independent of \(\varepsilon\)) and \(C_1\) can be taken arbitrarily large.
startsection section1@-3.5ex plus -1ex minus -.2ex2.3ex plus .2exStatement of the problem and regularised version The aim of this section is to introduce the proper framework to study the problem 1 , namely the Schrödinger ultrahyperbolic equations induced by an ultrahyperbolic operator \(A\) defined as in 8 . Subsequently, we study a regularised version of problem 1 and we prove Theorem 23 which will be fundamental for obtaining the existence and the uniqueness results.
We focus on singular operators, that can be formally written as \[\label{def46P} P=D_t-\sum_{i,j=1}^nD_{x_i}(a_{ij}(x)D_{x_j})- \sum_{k=1}^nb_k(x)D_{x_k}-V(x),\tag{21}\] where, in general, \(a_{ij},b_k,V \in \mathscr{D}'(\mathbb{R}^n)\), for all \(i,j,k=1,\dots,n\). As discussed in the previous section we assume that the net of regularised operators \((A_\varepsilon)_\varepsilon\), associated with the principal part of \(P\), defined by \[A_\varepsilon=\sum_{i,j=1}^nD_{x_i}(a_{ij,\varepsilon}(x)D_{x_j}), \quad \varepsilon \in (0,1],\] with \(a_{ij,\varepsilon}\) as in 10 , satisfies 1, 2 and 3. Furthermore, denoting by \((\varphi_{\omega(\varepsilon)})_{\varepsilon \in (0,1]}\) the family of mollifiers defined as in 7 , we define \[b_{k,\varepsilon}:=b_k \ast \varphi_{\omega(\varepsilon)}, \;\;k=1,\dots,n, \quad \quad V_\varepsilon:=V \ast \varphi_{\omega(\varepsilon)},\] and we work under the following additional hypotheses on the "first-order" and "zero-order" parts of the operator \(P\). We assume:
4. There exists a universal constant \(c_0>0\) such that, for each \(\varepsilon \in (0,1]\) (with \(N>1\) as in 3 and \(b_{k,\varepsilon}=b_k \ast \varphi_{\omega(\varepsilon)}\)), \[|\mathrm{Im}(b_{k,\varepsilon})(x)| \leq c_0 \langle x \rangle^{-N}, \quad \forall x \in \mathbb{R}^n, \;\forall k = 1, \dots, n,\] and there exists \(N_1 \in \mathbb{N}\), such that for each \(\beta \in \mathbb{N}_0^n\) and for each \(\varepsilon\in (0,1]\) \[\label{eq46bkeps} |\partial_x^\beta b_{k,\varepsilon}(x)|\leq C_\beta\omega(\varepsilon)^{-|\beta|-N_1}, \quad \forall x \in \mathbb{R}^n, \;\forall k = 1, \dots, n,\qquad{(15)}\] for some constant \(C_\beta>0\).
5. There exists \(N_2 \in \mathbb{N}\), such that for each \(\gamma \in \mathbb{N}_0^n\) and for each \(\varepsilon \in (0,1]\) (with \(V_\varepsilon=V \ast \varphi_{\omega(\varepsilon)}\)), \[\label{eq46Veps} |\partial_x^\gamma V_\varepsilon(x)| \leq C_\gamma \omega(\varepsilon)^{-|\gamma|-N_2}, \quad \forall x \in \mathbb{R}^n,\qquad{(16)}\] for some constant \(C_\gamma>0\)
We denote by \((P_\varepsilon)_\varepsilon\) the family of operators defined by \[\label{eq46Peps} \begin{align} P_\varepsilon&=D_t-A_\varepsilon-B_\varepsilon-V_\varepsilon\\ &:=D_t-\sum_{i,j=1}^nD_{x_i}(a_{ij,\varepsilon}(x)D_{x_j}) -\sum_{k=1}^nb_{k,\varepsilon}(x)D_{x_k}-V_\varepsilon(x), \quad \varepsilon \in (0,1]. \end{align}\tag{22}\] Note that \(B_\varepsilon=\mathrm{Op}(b_\varepsilon)\), where for each \(\varepsilon\in (0,1]\), with the same notation as above, one has \[\label{eq46beps} |\partial_x^\beta \partial_\xi^\alpha b_\varepsilon(x,\xi)|\leq C_{\alpha \beta}\omega(\varepsilon)^{-|\beta|}\langle \xi \rangle^{1-|\alpha|}, \quad \forall \alpha,\beta \in \mathbb{N}_0^n, \;(x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n.\tag{23}\] Henceforth, we say that \((P_\varepsilon)_\varepsilon\) satisfies condition 1, 2, 3, 4, 5, which means that the nets of coefficients of \((P_\varepsilon)_\varepsilon\) we consider satisfy such conditions.
Before discussing in detail the notion of weak solution for problems of the form 1 (i.e. the Cauchy problem induced by the operator \(P\) defined in 21 ), we prove the following theorem, which provides sufficient conditions for the singular operator \(P\) to generate a net of operators satisfying conditions 1-5.
Theorem 20. Let \(P\) be an operator of the form \[P=D_t-\sum_{i,j=1}^nD_{x_i}(a_{ij}(x)D_{x_j})- \sum_{k=1}^nb_k(x)D_{x_k}-V(x),\] and assume that the coefficients fulfill the following hypotheses.
The matrix \(\mathcal{A}(x)=(a_{ij}(x))_{i,j=1}^n\) is real and symmetric, for all \(x \in \mathbb{R}^n\) and its entries can be decomposed as \[a_{ij}(x)=c_{ij}+\tilde{a}_{ij}(x), \quad i,j=1,\dots,n,\] where \(c_{ij} \in \mathbb{R}\) and \(\tilde{a}_{ij} \in W^{1,\infty}(\mathbb{R}^n)\), for all \(i,j=1,\dots,n\). Moreover, we assume that the matrix \(\mathcal{C}=(c_{ij})_{i,j=1}^n\) is non-degenerate, namely there exists \(\mu'>0\) such that \[{\mu'}^{-1}|\xi|\leq |\mathcal{C} \cdot \xi |\leq \mu' |\xi|, \quad \forall \xi \in \mathbb{R}^n.\] Finally, the perturbations are assumed to have the form \[\label{eq46smalltildeaij} \tilde{a}_{ij}(x)=\langle x \rangle^{-N}\tilde{a}_{ij}'(x),\quad i,j=1,\dots,n,\qquad{(17)}\] where \(N \in \mathbb{N}\), \(N>1\) and \(\tilde{a}_{ij}' \in W^{1,\infty}(\mathbb{R}^n)\), such that \(\|\tilde{a}_{ij}'\|_{W^{1,\infty}}\leq \nu'\), for some \(\nu'>0\) sufficiently small, for all \(i,j=1,\dots,n\).
For all \(k=1,\dots,n\), we assume \(\mathrm{Re}(b_k) \in \mathscr{E}'(\mathbb{R}^n)\) and (with \(N>1\) as above) \[\mathrm{Im}(b_k)=\langle x \rangle^{-N}b_k',\] where \(\|b_{k}'\|_{L^\infty} \leq c_0'\), for some universal constant \(c_0'>0\).
\(V \in \mathscr{E}'(\mathbb{R}^n)\).
Then, the corresponding regularised operators \((P_\varepsilon)_\varepsilon\) defined as in Section [sec:sec46stateandreg], fulfills 1-5.
Proof. The hypothesis 1 easily follows from the fact that \(\mathcal{A}(x)\) is real and symmetric (for all \(x\)) and from the definition of the mollifier (cf. 7 ).
We now prove 2. Since \[a_{ij}=c_{ij}+\tilde{a}_{ij}, \quad i,j=1,\dots,n,\] denoting by \(\tilde{\mathcal{A}}_\varepsilon\) the matrix with entries \(\tilde{a}_{ij,\varepsilon}=\tilde{a}_{ij}\ast \varphi_{\omega(\varepsilon)}\), we have \[\label{eq46Aepssc} |\mathcal{A}_\varepsilon(x)\cdot \xi|=|(\mathcal{C}+\tilde{\mathcal{A}}_\varepsilon(x))\cdot \xi| \geq |\mathcal{C} \cdot \xi |-|\tilde{\mathcal{A}}_\varepsilon(x) \cdot \xi|\geq {\mu'}^{-1} |\xi|-|\tilde{\mathcal{A}}_\varepsilon(x) \cdot \xi|.\tag{24}\] Next, note that \[|\tilde{\mathcal{A}}_\varepsilon(x) \cdot \xi|^2=\sum_{i=1}^n\Bigl(\sum_{j=1}^n\tilde{a}_{ij,\varepsilon}(x)\xi_j\Bigr)^2\leq C \sum_{i=1}^n\sum_{j=1}^n\tilde{a}_{ij,\varepsilon}^2(x)\xi_j^2\leq C \max_{i,j}\|\tilde{a}_{ij,\varepsilon}\|^2_{L^\infty}|\xi|^2,\] for some \(C=C(n)>0\), and that, by ?? , \[\|\tilde{a}_{ij,\varepsilon}\|_{L^\infty} \leq \|\varphi_{\omega(\varepsilon)}\|_{L^1} \|\tilde{a}_{ij}\|_{L^\infty} \leq \nu',\quad \forall i,j=1,\dots,n.\] Therefore, choosing \(\nu'>0\) small enough, from 24 , we obtain \[|\mathcal{A}_\varepsilon(x)\cdot \xi|\geq {\mu'}^{-1} |\xi|-C\nu' |\xi|\geq \mu_1 |\xi|, \quad \forall \xi \in \mathbb{R}^n,\] for some \(\mu_1>0\) independent of \(\varepsilon\). By repeating the same argument we also have \[|\mathcal{A}_\varepsilon(x)\cdot \xi |\leq |\mathcal{C} \cdot \xi|+|\tilde{\mathcal{A}}_\varepsilon\cdot \xi|\leq \mu' |\xi|+C\nu' |\xi|\leq \mu_2|\xi|, \quad \forall \xi \in \mathbb{R}^n,\] for some \(\mu_2>0\) independent of \(\varepsilon\). Hence 2 holds.
The hypothesis 3 follows directly from Remark 14. As for 4 and 5 we have that, for all \(k=1,\dots, n\), \[\begin{align} |\mathrm{Im}(b_k) \ast \varphi_{\omega(\varepsilon)}(x)|&\leq \int_{\mathbb{R}^n}|\langle x-y \rangle^{-N}b_k'(x-y)\varphi_{\omega(\varepsilon)}(y)|dy\\ &\leq \langle x \rangle^{-N} \int |b_k'(x-y)\langle y \rangle^N \varphi_{\omega(\varepsilon)}(y)|dy\\ &\leq C\langle x \rangle^{-N}\|b_{k}'\|_{L^\infty}\\ &\leq c_0 \langle x \rangle^{-N}, \quad \forall x \in \mathbb{R}^n, \end{align}\] where the constants \(C,c_0>0\) depend on the mollifier but are independent of \(\varepsilon\). Finally, for all \(k=1,\dots,n\), since \(b_k \in \mathscr{E}'(\mathbb{R}^n)\), by Proposition 2.1 in [15], we get \[|\bigl(\partial_x^\alpha (b_k \ast \varphi_{\omega(\varepsilon)})\bigr)(x)|=|\bigl(b_k \ast \partial_x^\alpha \varphi_{\omega(\varepsilon)}\bigr)(x)|\leq C_{\alpha}\omega(\varepsilon)^{-|\alpha|-N_1}, \quad \forall x \in \mathbb{R}^n,\] and analogously \[|\bigl(\partial_x^\alpha (V \ast \varphi_{\omega(\varepsilon)})\bigr)(x)|=|\bigl(V \ast \partial_x^\alpha \varphi_{\omega(\varepsilon)}\bigr)(x)|\leq C_{\alpha}'\omega(\varepsilon)^{-|\alpha|-N_2}, \quad \forall x \in \mathbb{R}^n,\] for all \(\alpha \in \mathbb{N}^n\) and for \(C_\alpha,C_\alpha'>0\), \(N_1=N_1(n),N_2=N_2(n)\), independent of \(\varepsilon\). Therefore 4 and 5 hold and the proof ends. ◻
Remark 21. Note that, in Theorem 20, it would be possible to increase the magnitude of the perturbations \(\tilde{a}_{ij}\) and require an \(L^\infty\)-bound (depending on the size of the constant coefficients \(c_{ij}\)) given by a larger constant than the \(\nu\) used for the derivatives (cf. Remark 32). However, to keep the presentation as clear as possible, we have chosen to require the coefficients and their derivatives to be bounded by the same constant in the \(L^\infty\)-norm.
Our goal now is to find an \(H^\infty\)-weak solution of problem 1 , which is based on the notion of moderate net of functions as stated in Subsection 0.3 in the following sense. For the sake of completeness we state also the notion of \(H^s\)-weak solution that will be useful for some class of problems (see Remark 26 below).
Definition 22. The net \((u_\varepsilon)_\varepsilon \in \lbrace C([0,T],H^\infty(\mathbb{R}^n))\rbrace^{(0,1]}\) (resp. \((u_\varepsilon)_\varepsilon \in \lbrace C([0,T],H^s(\mathbb{R}^n))\rbrace^{(0,1]}\)) is an \(H^\infty\)-weak solution (resp. \(H^s\)-weak solution)* of 1 , if there exist \((u_{0,\varepsilon})_{\varepsilon}\) and \((f_\varepsilon)_\varepsilon\) \(H^\infty\)-moderate (resp. \(H^s\)-moderate) regularisations of \(u_0\) and \(f\), respectively, such that, for any \(\varepsilon \in (0,1]\), \(u_\varepsilon\) solves \[\label{probveryweak} \begin{cases} P_\varepsilon u_\varepsilon=f_\varepsilon \quad \text{in} \;\;(0,T]\times \mathbb{R}^n, \\ u_\varepsilon(0,\cdot)=u_{0,\varepsilon} \quad \text{in} \;\;\mathbb{R}^n, \end{cases}\tag{25}\] and \((u_\varepsilon)_\varepsilon\) is \(H^\infty\)-moderate (resp. \(H^s\)-moderate).*
In order to study the weak well-posedness of problem 1 , the first step is to consider, for fixed \(\varepsilon\in (0,1]\), the \(\varepsilon\)-regularised problem (cf. [15]) \[\label{eq46reg} \begin{cases} P_\varepsilon u=g \quad \text{in} \;\;(0,T]\times \mathbb{R}^n, \\ u_\varepsilon(0,\cdot)=v_0 \quad \text{in} \;\;\mathbb{R}^n, \end{cases}\tag{26}\] where \(v_0 \in H^s(\mathbb{R}^n)\) and \(g\) has suitable regularity.
The key result to obtain the existence and uniqueness of an \(H^\infty\)-weak solution is Theorem 23 below. Indeed, the estimates we derive in this theorem for the solution of 26 (for each fixed \(\varepsilon\)) ensure the moderateness of the net \((u_\varepsilon)_\varepsilon\), as well as the uniqueness (in the sense explained in Subsection 0.9 below) of an \(H^\infty\)-weak solution, which we discuss in detail in Section [sec:sec46exandun].
The proof of Theorem 23 follows the same pattern as Lemma 4.1 in [4] (cf. [5], [19]). We refer the reader to [4] for further details.
Theorem 23. Let \(\varepsilon \in (0,1]\) and consider the regularised problem 26 with Cauchy data \(v_0 \in H^s(\mathbb{R}^n)\) and assume that 1-5 hold for \((P_\varepsilon)_\varepsilon\).
Then, we have:
If \(g\in L^1([0,T];H^s(\mathbb{R}^n))\), the IVP 26 has a unique solution \(u_\varepsilon\in C([0,T];H^s(\mathbb{R}^n))\) satisfying \[\label{eq46i} \sup_{0\leq t\leq T}\|u_\varepsilon(t,\cdot)\|_s\leq C_2e^{C_1\omega(\varepsilon)^{-k_1}T}\left(\|v_0\|_s+\int_0^T \|g(t,\cdot)\|_sdt\right).\qquad{(18)}\]
If \(g\in L^2([0,T];H^s(\mathbb{R}^n))\), the IVP 26 has a unique solution \(u_\varepsilon\in C([0,T];H^{s}(\mathbb{R}^n))\) satisfying \[\begin{align} &\sup_{0\leq t \leq T}\|u_\varepsilon(t,\cdot)\|^2_s+\omega(\varepsilon)^{-k}\int_0^T \|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u_\varepsilon(t,\cdot)\|_0^2\,dt\\ & \leq C_2e^{C_1\omega(\varepsilon)^{-k_1}T}\left(\|v_0\|_s^2+\int_0^T \|g(t,\cdot)\|_s^2 dt\right). \end{align}\]
If \(\Lambda^{s-\frac{m-1}{2}}g\in L^2([0,T]\times \mathbb{R}^n;\langle x \rangle^N dx dt)\), the IVP 26 has a unique solution \(u_\varepsilon\in C([0,T];H^{s}(\mathbb{R}^n))\) satisfying \[\begin{align} &\sup_{0\leq t \leq T}\|u_\varepsilon(t,\cdot)\|^2_s+\omega(\varepsilon)^{-k} \int_0^T \|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u_\varepsilon(t,\cdot)\|_0^2\,dt\\ &\leq C_2e^{C_1\omega(\varepsilon)^{-k_1}T}\left(\|v_0\|_s^2+\int_0^T \| \langle x \rangle^{N/2}\Lambda^{s-1/2}g(t,\cdot)\|^2_0 \,dt\right), \end{align}\]
where \(C_1,C_2>0\) and \(k, k_1 \in \mathbb{N}\), depend on the coefficient of \(P\) defined in 22 and on \(s\), but are independent of \(\varepsilon\).
Proof. The idea of the proof, following the approach in [4], is to define, for each fixed \(\varepsilon\), a norm \(N^s_\varepsilon(u)\) on the Sobolev space \(H^s(\mathbb{R}^n)\) that is equivalent to the standard norm \(\|\cdot\|_s\) (see [4]), and then to estimate the time derivative of this norm, keeping track of the explicit dependence on \(\varepsilon\).
To this end, fix \(\varepsilon \in (0,1]\) and \(s \in \mathbb{R}\). Define, for \(\varepsilon \in (0,1]\) the operator \(E_\varepsilon=\mathrm{Op}(e^{d_\varepsilon})\in \Psi^0\), where \(d_\varepsilon\in S^0\) is the symbol constructed in Lemma 17 satisfying ?? . We then introduce the quantity \[N^s_\varepsilon(u):=\bigl(\|E_\varepsilon \Lambda^su\|_0^2+\|u\|_{s-1}^2)^{1/2}.\] It is also convenient to denote \(\tilde{E}_\varepsilon=\mathrm{Op}(e^{-d_\varepsilon})\) and write \(\tilde{E}_\varepsilon E_\varepsilon=I+R_{-1,\varepsilon}\), with \(R_{-1,\varepsilon}=\mathrm{Op}(r_{-1,\varepsilon})\), \(r_{-1,\varepsilon} \in S^{-1}\). By Theorem 1 and Theorem 3 we get \[\label{eq46normeq1} \begin{align} \|u\|_s^2 &\leq \|\tilde{E}_\varepsilon E_\varepsilon\Lambda^su\|_0^2+\|R_{-1,\varepsilon}\Lambda^su\|_0^2\\ &\leq C_1'(|e^{-d_\varepsilon}|_{k_1'}^{(0)}\|E_\varepsilon \Lambda^s u\|_0^2+|r_{-1,\varepsilon}|_{k_1''}^{(-1)}\|u\|_{s-1}^2) \\ &\leq C_1''\Bigl(|e^{-d_\varepsilon}|_{k_1'}^{(0)}+|e^{d_\varepsilon}|_{k_1'''}^{(0)}|e^{-d_\varepsilon}|^{(0)}_{k_1'''}\Bigr)N^s_\varepsilon(u)^2\\ &\leq C_1 \omega(\varepsilon)^{-k_1}N_\varepsilon^s(u)^2, \quad u \in H^s(\mathbb{R}^n), \end{align}\tag{27}\] where \(C_1,C_1',C_1''>0\) \(k_1',k_1'',k_1''' \in \mathbb{N}_0\), depend on \(s\) and on the dimension \(n\), but are independent of \(\varepsilon\), and \(k_1=\max \lbrace k_1'-1,2k_1'''-2 \rbrace\). Similarly, \[\label{eq46normeq2} \begin{align} N_\varepsilon(u)^2 &\leq (C_2'|e^{d_\varepsilon}|_{k_2'}^{(0)}+1)\|u\|_s^2 \\ &\leq C_2\omega(\varepsilon)^{-k_2}\|u\|_s^2, \quad u \in H^s(\mathbb{R}^n), \end{align}\tag{28}\] with \(C_2,C_2'>0\), \(k_2' \in \mathbb{N}_0\), independent of \(\varepsilon\) (but, as before depend on \(s\) and on \(n\)) and \(k_2=k_2'-1\). Therefore, 27 and 28 show that the two norms \(N_\varepsilon^s(\cdot)\), and \(\|\cdot \|_s\) are indeed equivalent, and provide precise information on the dependence of equivalent constants on \(\varepsilon\).
Let now \(u_\varepsilon \in C([0,T];H^s(\mathbb{R}^n))\) be the solution of 26 (in the case of \(g \in L^1([0,T],H^s(\mathbb{R}^n)\)). For the existence of such a \(u_\varepsilon\) we refer to [4] (cf. [5] and Remark 29 below). Our aim is to estimate \(\partial_t N_\varepsilon^s(u_\varepsilon)^2\). By integrating in time, and using the equivalence shown in 27 and 28 , this estimate will lead to the desired estimate \((i)\), \((ii)\) and \((iii)\).
To do so, we start by examining the term \(\partial_t \|u_\varepsilon\|_{s-1}^2\). By using the self-adjoint property of \(A_\varepsilon\), and once again Theorem 1 and Theorem 3, we get \[\begin{align} \partial_t \|u_\varepsilon\|_{s-1}^2&=2\mathrm{Re}(\Lambda^{s-1}\partial_tu_\varepsilon,\Lambda^{s-1}u_\varepsilon)_0 \\ &=2\mathrm{Re}(\Lambda^{s-1}(i(A_\varepsilon+B_\varepsilon+V_\varepsilon)u_\varepsilon+ig),\Lambda^{s-1}u_\varepsilon)_0 \\ &=2\mathrm{Re}(i\Lambda^{s-1}A_\varepsilon u_\varepsilon,\Lambda^{s-1}u_\varepsilon)_0+2\mathrm{Re}(i\Lambda^{s-1}(B_\varepsilon+V_\varepsilon)u_\varepsilon,\Lambda^{s-1}u_\varepsilon)_0+2\mathrm{Re}(i\Lambda^{s-1}g,\Lambda^{s-1}u_\varepsilon)_0 \\ &=2 \mathrm{Re}(i[\Lambda^{s-1},A_\varepsilon] u_\varepsilon,\Lambda^{s-1}u_\varepsilon)_0+2\mathrm{Re}(i\Lambda^{s-1}(B_\varepsilon+V_\varepsilon)u_\varepsilon,\Lambda^{s-1}u_\varepsilon)_0+2\mathrm{Re}(i\Lambda^{s-1}g,\Lambda^{s-1}u_\varepsilon)_0 \\ & \leq C'\Bigl(\bigl(|a_\varepsilon|_{k'}^{(2)}+|b_\varepsilon|^{(1)}_{k''}+|V_\varepsilon|^{(0)}_{k'''}\bigr)\|u_\varepsilon\|_s^2+\|g\|_s\|u_\varepsilon\|_s\Bigr). \end{align}\] Therefore, by 14 , 13 , ?? , 23 , we obtain \[\label{eq46smeno1} \partial_t \|u_\varepsilon\|_{s-1}^2 \leq C \Bigl(\omega(\varepsilon)^{-k}\|u_\varepsilon\|_s^2+\|g\|_s\|u_\varepsilon\|_s\Bigr),\tag{29}\] for some \(k=k(n,s)\) and some \(C=C(n,s)>0\) independent of \(\varepsilon\). We next estimate the term \(\partial_t\|E_\varepsilon\Lambda^su_\varepsilon\|_0^2\). By using again the self-adjoint property of \(A_\varepsilon\) we have \[\label{eq461piu2piu3} \begin{align} \partial_t\|E_\varepsilon\Lambda^su_\varepsilon\|_0^2&=2\mathrm{Re}(E_\varepsilon\Lambda^s\partial_tu_\varepsilon,E_\varepsilon\Lambda^su_\varepsilon)_0 \\ &\leq 2\mathrm{Re}(E_\varepsilon\Lambda^s iA_\varepsilon u_\varepsilon,E_\varepsilon\Lambda^s u_\varepsilon)_0+2\mathrm{Re}(E_\varepsilon\Lambda^s i(B_\varepsilon+V_\varepsilon) u_\varepsilon,E_\varepsilon\Lambda^su_\varepsilon)_0\\ &\quad + 2\mathrm{Re}(E_\varepsilon\Lambda^s ig,E_\varepsilon\Lambda^su_\varepsilon)_0\\ &\leq 2\mathrm{Re}([E_\varepsilon\Lambda^s, iA_\varepsilon] u_\varepsilon,E_\varepsilon\Lambda^s u_\varepsilon)_0+2\mathrm{Re}(E_\varepsilon\Lambda^s i(B_\varepsilon+V_\varepsilon) u_\varepsilon,E_\varepsilon\Lambda^su_\varepsilon)_0\\ &\quad + 2\mathrm{Re}(E_\varepsilon\Lambda^s ig,E_\varepsilon\Lambda^su_\varepsilon)_0\\ &=(I)+(II)+(III). \end{align}\tag{30}\] Hence, to estimate \(\partial_t\|E_\varepsilon\Lambda^su_\varepsilon\|_0^2\) we estimate \((I)\), \((II)\) and \((III)\) separately.
For \((I)\), as in [4], we have \[(I)=2\mathrm{Re}([E_\varepsilon,iA_\varepsilon]\Lambda^su_\varepsilon,E_\varepsilon\Lambda^s u_\varepsilon)_0+2\mathrm{Re}(E_\varepsilon[\Lambda^s,iA_\varepsilon]u_\varepsilon,E_\varepsilon\Lambda^su_\varepsilon)_0.\] At this point, we note that, by Theorem 3 \[\begin{align} [E_\varepsilon,iA_\varepsilon]&=\mathrm{Op}(\lbrace e^{d_\varepsilon},a_\varepsilon\rbrace)+\mathrm{Op}(r^1_{0,\varepsilon})\\ &=\mathrm{Op}(\lbrace d_\varepsilon,a_\varepsilon\rbrace e^{d_\varepsilon})+\mathrm{Op}(r^1_{0,\varepsilon})\\ &=\mathrm{Op}(-\lbrace a_\varepsilon, d_\varepsilon\rbrace)E_\varepsilon+\mathrm{Op}(r^2_{0,\varepsilon}), \end{align}\] and for all \(k \in \mathbb{N}_0\), there exist \(k_1 \in \mathbb{N}_0\) and \(C=C(k)>0\), such that \[|r^1_{0,\varepsilon}|_k^{(0)}+|r^2_{0,\varepsilon}|_{k}^{(0)} \leq C |a_\varepsilon|_{k_1}^{(2)}\Bigl( |d_\varepsilon|_{k_1}^{(0)}+(|d_\varepsilon|_{k_1}^{(0)})^2\Bigr).\] Moreover, \[\begin{align} E_\varepsilon[\Lambda^s,iA_\varepsilon]&=E_\varepsilon[\Lambda^s,iA_\varepsilon]\Lambda^{-s}\tilde{E}_\varepsilon E_\varepsilon\Lambda^s+\mathrm{Op}(r^3_{0,\varepsilon})\\ &=\mathrm{Op}(\lbrace \langle \xi \rangle^{s},a_\varepsilon\rbrace \langle \xi \rangle^{-s})E_\varepsilon\Lambda^s+\mathrm{Op}(r_{s,\varepsilon}^1) \end{align}\] where, once again, for any \(k \in \mathbb{N}_0\), there exist \(k_1' \in \mathbb{N}_0\) and \(C'=C'(k)>0\), such that \[|r^3_{0,\varepsilon}|_k^{(0)}+|r^1_{s,\varepsilon}|_k^{(s)} \leq C' |a_\varepsilon|_{k_1'}^{(2)}|d_\varepsilon|_{k_1'}^{(0)}.\] Thus (recall \(H_{a_\varepsilon}d_\varepsilon=\lbrace a_\varepsilon, d_\varepsilon\rbrace\)) \[\begin{align} (I)&= -2\mathrm{Re}(\mathrm{Op}(H_{a_\varepsilon}d_\varepsilon-\lbrace \langle \xi \rangle^{s},a_\varepsilon\rbrace \langle \xi \rangle^{-s})E_\varepsilon\Lambda^su_\varepsilon,E_\varepsilon\Lambda^su_\varepsilon)_0+2\mathrm{Re}(\mathrm{Op}(r_{s,\varepsilon}^2)u_\varepsilon,E_\varepsilon\Lambda^su_\varepsilon)_0 \end{align}\] with \(r_{s,\varepsilon}^2 \in S^s\) having seminorms bounded by the seminorms of \(a_\varepsilon\) and \(d_\varepsilon\), as before.
For \((II)\) we have \[(II)=2\mathrm{Re}(i(B_\varepsilon+V_\varepsilon)E_\varepsilon\Lambda^su_\varepsilon,E_\varepsilon\Lambda^su_\varepsilon)_0+(\mathrm{Op}(r^3_{s,\varepsilon})u_\varepsilon,E_\varepsilon\Lambda^s u_\varepsilon)_0,\] and then \[\label{eq461piu2} \begin{align} (I)+(II)&=2\mathrm{Re}(\mathrm{Op}(-H_{a_\varepsilon}d_\varepsilon+\lbrace \langle \xi \rangle^{s},a_\varepsilon\rbrace \langle \xi \rangle^{-s}+ib_\varepsilon)E_\varepsilon\Lambda^su_\varepsilon,E_\varepsilon\Lambda^su_\varepsilon)_0 \\ &\quad +2\mathrm{Re}(iV_\varepsilon E_\varepsilon\Lambda^su_\varepsilon,E_\varepsilon\Lambda^su_\varepsilon)_0+(\mathrm{Op}(r^2_{s,\varepsilon}+r^3_{s,\varepsilon})u_\varepsilon,E_\varepsilon\Lambda^s u_\varepsilon)_0. \end{align}\tag{31}\] The idea now is to use the Sharp Gårding inequality (see Theorem 4) to estimate the first term in 31 . To this end, we use the hypotheses 2, 4 and ?? to obtain (recall \(B_\varepsilon=\mathrm{Op}(b_\varepsilon)\)) \[\begin{align} \mathrm{Re}\Bigl(-H_{a_\varepsilon}d_\varepsilon+\lbrace \langle \xi \rangle^{s},a_\varepsilon\rbrace \langle \xi \rangle^{-s}+ib_\varepsilon\Bigr)&=-H_{a_\varepsilon}d_\varepsilon+\lbrace \langle \xi \rangle^{s},a_\varepsilon\rbrace \langle \xi \rangle^{-s}-\mathrm{Im}(b_\varepsilon)\\ &\leq -C_1\langle x \rangle^{-N}|\xi|+C_2+C_3\nu \langle x\rangle^{-N}|\xi|+c_0\langle x \rangle^{-N}|\xi|, \end{align}\] with \(C_2,C_3,\nu,c_0>0\) universal constants and \(C_1>0\) can be taken arbitrarily large (see Remark 19). Hence, by using \((1+|\xi|^2)^{1/2}\leq 1+|\xi|\), \[\mathrm{Re}\Bigl(-H_{a_\varepsilon}d_\varepsilon+\lbrace \langle \xi \rangle^{s},a_\varepsilon\rbrace \langle \xi \rangle^{-s}+ib_\varepsilon\Bigr) \leq -C\langle x \rangle^{-N}\langle\xi\rangle+C_2',\] with \(C,C_2'>0\) universal constants independent of \(\varepsilon\). Thus, by Theorem 4, we get \[\begin{align} (I)+(II) &\leq -2C\mathrm{Re}(\langle x \rangle^{-N}\Lambda E_\varepsilon\Lambda^{s} u_\varepsilon,E_\varepsilon\Lambda^s u_\varepsilon)_0+C_2'\|E_\varepsilon\Lambda^su_\varepsilon\|_0^2 \\ &\quad +2\mathrm{Re}(iV_\varepsilon E_\varepsilon\Lambda^su_\varepsilon,E_\varepsilon\Lambda^su_\varepsilon)_0+(\mathrm{Op}(r^2_{s,\varepsilon}+r^3_{s,\varepsilon})u_\varepsilon,E_\varepsilon\Lambda^s u_\varepsilon)_0 \\ &\leq -2C\mathrm{Re}(\langle x \rangle^{-N}\Lambda E_\varepsilon\Lambda^{s} u_\varepsilon,E_\varepsilon\Lambda^s u_\varepsilon)_0+C_1 \omega(\varepsilon)^{-k_1}N_\varepsilon^s(u_\varepsilon)^2, \end{align}\] for some \(C,C_1>0\) and \(k_1 \in \mathbb{N}\), possibly differing from previous occurrences but independent of \(\varepsilon\). We now note that (cf. [5], p. 393) \[\langle x \rangle^{-N} \Lambda=\mathrm{Op}(\langle x \rangle^{-N}\langle \xi \rangle)=\mathrm{Op}(\langle x \rangle^{-N/2}\langle \xi \rangle^{1/2})\mathrm{Op}(\langle x \rangle^{-N/2}\langle \xi \rangle^{1/2})+\mathrm{Op}(r_0), \quad r_0 \in S^0,\] and, by Theorem 2, \[\mathrm{Op}(\langle x \rangle^{-N/2}\langle \xi \rangle^{1/2})=\bigl(\mathrm{Op}(\langle x \rangle^{-N/2}\langle \xi \rangle^{1/2})\bigr)^\ast + \mathrm{Op}(r_{-1/2}), \quad r_{-1/2} \in S^{-1/2}.\] Thus, possibly enlarging the universal constant \(C_1\), we get \[\label{eq46quasifinalIandII} (I)+(II) \leq -2C\mathrm{Re}(\langle x \rangle^{-N/2}\Lambda^{1/2} E_\varepsilon\Lambda^{s} u_\varepsilon,\langle x \rangle^{-N/2}\Lambda^{1/2}E_\varepsilon\Lambda^s u_\varepsilon)_0+C_1 \omega(\varepsilon)^{-k_1}N_\varepsilon^s(u_\varepsilon)^2.\tag{32}\] Now, note that by using once again the properties of the calculus stated in Theorem 1 and Theorem 3, we get \[\label{eq46us} \begin{align} \|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u_\varepsilon\|_0&=\|\langle x \rangle^{-N/2}\Lambda^{1/2}\tilde{E}_\varepsilon E_\varepsilon\Lambda^s u_\varepsilon\|_0+\|\langle x\rangle^{-N/2}\Lambda^{1/2}R_{-1,\varepsilon}\Lambda^{s}u_\varepsilon\|_0\\ &\leq C'\omega(\varepsilon)^{-k'}\|\langle x \rangle^{-N/2}\Lambda^{1/2}E_\varepsilon\Lambda^s u_\varepsilon\|_0+C''\omega(\varepsilon)^{-k''}N_\varepsilon^s(u_\varepsilon), \end{align}\tag{33}\] for some universal constants \(C',C''>0\) and \(k',k'' \in \mathbb{N}\).
Therefore, by combining 32 and 33 with possibly different constants \(C,C_1>0\) and \(k,k_1 \in \mathbb{N}\), we obtain \[\label{eq46fin1piu2} (I)+(II) \leq -C\omega(\varepsilon)^{-k}\|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u_\varepsilon\|_0^2+C_1\omega(\varepsilon)^{-k_1}N_\varepsilon^s(u_\varepsilon)^2.\tag{34}\] The estimate of the term \((III)\) in 30 depends on which of the estimate \((i)\), \((ii)\), or \((iii)\) in the statement we aim to obtain. In what follows, we denote by \(C,C_1,C_2>0\) and \(k,k_1,k_2 \in \mathbb{N}\) some constants, possibly changing from line to line, but always independent of \(\varepsilon\).
Case \((i)\). In this case it is sufficient to note that \[\label{eq46est3} (III)=2\mathrm{Re}(E_\varepsilon\Lambda^s ig,E_\varepsilon\Lambda^su_\varepsilon)_0 \leq C_2\omega(\varepsilon)^{-k_2}N_\varepsilon^s(g)N_\varepsilon^s(u_\varepsilon).\tag{35}\] Hence, by 29 , 30 and 34 , we get \[\partial_t N^s_\varepsilon(u_\varepsilon)^2 \leq C_1\omega(\varepsilon)^{-k_1}N_\varepsilon^s(u_\varepsilon)^2+C_2\omega(\varepsilon)^{-k_2}N_\varepsilon^s(g)N_\varepsilon^s(u_\varepsilon)\] and so \[\partial_t N^s_\varepsilon(u_\varepsilon) \leq C_1\omega(\varepsilon)^{-k_1}N_\varepsilon^s(u_\varepsilon)+C_2\omega(\varepsilon)^{-k_2}N_\varepsilon^s(g),\] that we may rewrite as \[\partial_t\Bigl(e^{-C_1\omega(\varepsilon)^{-k_1}t}N_\varepsilon^s(u_\varepsilon)\Bigr)\leq e^{-C_1\omega(\varepsilon)^{-k_1}t}C_2\omega(\varepsilon)^{-k_2}N_\varepsilon^s(g).\] Therefore, by integrating in time and using the equivalence of the Sobolev norms \(N_\varepsilon^s(\cdot)\) and \(\|\cdot \|_s\) proved in 27 and 28 ), we obtain that, for each \(t \in [0,T]\), \[\begin{align} \|u_\varepsilon(t,\cdot)\|_s&\leq C_2\omega(\varepsilon)^{-k_2}e^{C_1\omega(\varepsilon)^{-k_1}T}\Bigl(\|v_0\|_s+\int_0^T \|g(t,\cdot)\|_s dt\Bigr)\\ &\leq C_2e^{C_1\omega(\varepsilon)^{-k_1}T}\Bigl(\|v_0\|_s+\int_0^T \|g(t,\cdot)\|_s dt\Bigr). \end{align}\]
Case \((ii)\). To get \((ii)\) we use again 29 , 34 and 35 , to obtain \[\label{eq46usest2} \begin{align} &\partial_t \Bigl(e^{-C_1\omega(\varepsilon)^{-k_1}t} N_\varepsilon^s(u_\varepsilon)^2 \Bigr)\\ &\quad \leq e^{-C_1\omega(\varepsilon)^{-k_1}t }\Bigl( -C\omega(\varepsilon)^{-k}\|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u_\varepsilon\|_0^2+C_2\omega(\varepsilon)^{-k_2}N_\varepsilon^s(g)N_\varepsilon^s(u_\varepsilon)\Bigr). \end{align}\tag{36}\] Moreover, note that \[\label{eq46useful} N_\varepsilon^s(g)N_\varepsilon^s(u_\varepsilon)\leq \omega(\varepsilon)N_\varepsilon^s(u_\varepsilon)^2+\omega(\varepsilon)^{-1}N_\varepsilon^s(g)^2.\tag{37}\] Hence, by reasoning as before, for each \(t \in [0,T]\) we get \[\begin{align} &\|u_\varepsilon(t,\cdot)\|^2_s+\omega(\varepsilon)^{-k}\int_0^T \|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u_\varepsilon(t,\cdot)\|_0^2\,dt\\ & \leq C_2\omega(\varepsilon)^{-k_2}e^{C_1\omega(\varepsilon)^{-k_1}T}\left(\|v_0\|_s^2+\int_0^T \|g(t,\cdot)\|_s^2 dt\right)\\ & \|u_\varepsilon(t,\cdot)\|^2_s+\omega(\varepsilon)^{-k}\int_0^T \|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u_\varepsilon(t,\cdot)\|_0^2\,dt\\ & \leq C_2e^{C_1\omega(\varepsilon)^{-k_1}T}\left(\|v_0\|_s^2+\int_0^T \|g(t,\cdot)\|_s^2 dt\right)\\ \end{align}\] Case \((iii)\). To obtain \((iii)\) we have to refine the estimate of the term \((III)\) in the following sense. By using again 37 , we have (cf. [4]) \[\begin{align} 2\mathrm{Re}(E_\varepsilon\Lambda^s ig,E_\varepsilon\Lambda^su_\varepsilon)_0&= 2\mathrm{Re}(\Lambda^{1/2}E_\varepsilon\Lambda^{s-1/2} ig,E_\varepsilon\Lambda^su_\varepsilon)_0+2\mathrm{Re}([E_\varepsilon,\Lambda^{1/2}] \Lambda^{s-1/2} ig,E_\varepsilon\Lambda^su_\varepsilon)_0\\ &\leq 2\mathrm{Re}(E_\varepsilon\Lambda^{s-1/2} ig,E_\varepsilon\Lambda^{s+1/2}u_\varepsilon)_0 + 2\mathrm{Re}(E_\varepsilon\Lambda^{s-1/2} ig,[\Lambda^{1/2},E]\Lambda^su_\varepsilon)_0\\ &\quad + 2\mathrm{Re}([E_\varepsilon,\Lambda^{1/2}] \Lambda^{s-1/2} g,E_\varepsilon\Lambda^su_\varepsilon)_0 \\ & \leq 2\mathrm{Re}(\langle x \rangle^{N/2}E_\varepsilon\Lambda^{s-1/2} ig,\langle x \rangle^{-N/2}E_\varepsilon\Lambda^{s+1/2}u_\varepsilon)_0 \\ &\quad + 2\mathrm{Re}( \langle x \rangle^{N/2}E_\varepsilon\Lambda^{s-1/2} ig,\langle x\rangle^{-N/2}[\Lambda^{1/2},E]\Lambda^su_\varepsilon)_0\\ &\quad + 2\mathrm{Re}(\langle x \rangle^{N/2}[E_\varepsilon,\Lambda^{1/2}] \Lambda^{s-1/2} g,\langle x \rangle^{-N/2}E_\varepsilon\Lambda^su_\varepsilon)_0 \\ &\leq C\omega(\varepsilon)^{-k}\|\langle x \rangle^{N/2}\Lambda^{s-1/2}g\|_0^2+\omega(\varepsilon)\Bigl(\|\langle x \rangle^{-N/2}\Lambda^{s+1/2}u_\varepsilon\|_0^2+ N_\varepsilon^s(u_\varepsilon)^2 \Bigr). \end{align}\] Therefore, in this case \[\partial_t N_\varepsilon^s(u_\varepsilon)^2 \leq -C\omega(\varepsilon)^{-k}\|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u_\varepsilon\|_0^2+C_1\omega(\varepsilon)^{-k_1}N_\varepsilon^s(u_\varepsilon)^2+C_2\omega(\varepsilon)^{-k_2}\|\langle x \rangle^{N/2}\Lambda^{s-1/2}g\|_0^2,\] which leads (by repeating the same argument of Cases \((i)\) and \((ii)\)) to the fact that, for all \(t \in [0,T]\), we have \[\begin{align} &\|u_\varepsilon(t,\cdot)\|^2_s+\omega(\varepsilon)^{-k} \int_0^T \|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u_\varepsilon(t,\cdot)\|_0^2\,dt\\ &\leq C_2\omega(\varepsilon)^{-k_2}e^{C_1\omega(\varepsilon)^{-k_1}T}\left(\|v_0\|_s^2+\int_0^T \| \langle x \rangle^{N/2}\Lambda^{s-1/2}g(t,\cdot)\|^2_0 \,dt\right)\\ &\leq C_2e^{C_1\omega(\varepsilon)^{-k_1}T}\left(\|v_0\|_s^2+\int_0^T \| \langle x \rangle^{N/2}\Lambda^{s-1/2}g(t,\cdot)\|^2_0 \,dt\right) \end{align}\] ◻
We are now ready to employ this theorem to prove the existence of an \(H^\infty\)-weak solution of the Cauchy problem 1 .
startsection section1@-3.5ex plus -1ex minus -.2ex2.3ex plus .2exExistence and uniqueness of an \(H^\infty\)-very weak solution The aim of this section is first to prove the existence of an \(H^\infty\)-very weak solution in the sense of Definition 22 and then to establish a uniqueness result for such a solution. As anticipated above, both results follow from the estimates obtained in Theorem 23. Note that when we choose singular initial data and right-hand side their regularisation naturally leads to a \(H^\infty\)-very weak solution. If we choose instead initial data and right-hand side in \(H^s(\mathbb{R}^n)\) and we do not regularise them, then we will generate an \(H^s\)-very weak solution (see Remark 26 below).
We start by studying the existence (recall Definition 22). The main result of this work is the following theorem, which, once again, relies on the estimates obtained in Theorem 23.
Theorem 24. Let the regularisation \((P_\varepsilon)_\varepsilon\) of the operator \(P\) in 1 fulfill the hypotheses 1-5 and let \(f \in C([0,T];H^{-\infty}(\mathbb{R}^n))\) and \(u_0 \in H^{-\infty}(\mathbb{R}^{n})\). Then there exists an \(H^\infty\)-very weak solution \((u_\varepsilon)_\varepsilon\) of the problem 1 .
Proof. Let \(\varphi \in \mathscr{S}(\mathbb{R}^n)\), with \(\int \varphi \, dx=1\). For \(\varepsilon\in (0,1]\) denote by \(f_\varepsilon(t,x)=f_\varepsilon\ast \varphi_\varepsilon\) and \(u_{0,\varepsilon}=u_0 \ast\varphi_\varepsilon\). If \(f \in C([0,T];H^{-\infty}(\mathbb{R}^n))\) and \(u_0 \in H^{-\infty}(\mathbb{R}^n)\) we know (see Remark 9) that for each \(s \in \mathbb{R}\) there exist \(N_f,N_{u_0} \in \mathbb{N}\) and \(C>0\) such that \[\|f_\varepsilon\|_s\leq C \varepsilon^{-N_f}, \quad \|u_{0,\varepsilon}\|_s \leq C \varepsilon^{-N_{u_0}},\] uniformly in \(t \in [0,T]\) and \(\varepsilon\in (0,1]\). Hence, by ?? , \[\begin{align} \sup_{0\leq t\leq T}\|u_\varepsilon(t,\cdot)\|_s&\leq C_2e^{TC_1\omega(\varepsilon)^{-k_1}}\left(\|u_{0,\varepsilon}\|_s+\int_0^T \|f_\varepsilon(t,\cdot)\|_sdt\right)\\ &\leq C_2'e^{TC_1\omega(\varepsilon)^{-k_1}}\varepsilon^{-N_f-N_{u_0}}. \end{align}\] Therefore, using the positive scale \((\omega_\varepsilon)_\varepsilon\) defined by \[\omega(\varepsilon):=\log(\log(\varepsilon))^{-1}, \quad \varepsilon\in (0,1],\] we have the \(H^\infty\)-moderateness of \((u_\varepsilon)_\varepsilon\), and then the existence of an \(H^\infty\)-weak solution, follows. Note that the scale \(\omega\) has been chosen to be independent of \(k_1\) and therefore independent of the Sobolev order \(s\), leading to \(H^\infty\)-moderateness. ◻
Remark 25. Note that, under the assumptions of Theorem 24, we have that the net of solutions \((u_\varepsilon)_\varepsilon\) generated, for each fixed \(s \in \mathbb{R}\), satisfies the following estimates with respect to the nets \((f_\varepsilon)_\varepsilon, (u_{0,\varepsilon})_{\varepsilon}\) associated with the Cauchy data \(f,u_0\) (see the proof of Theorem 24).
If \((f_\varepsilon)_\varepsilon\in \lbrace L^1([0,T];H^s(\mathbb{R}^n))\rbrace^{(0,1]}\) \[\sup_{0\leq t\leq T}\|u_\varepsilon(t,\cdot)\|_s\leq Ce^{C_1\omega(\varepsilon)^{-k_1}T}\left(\|u_{0,\varepsilon}\|_s+\int_0^T \|f_\varepsilon(t,\cdot)\|_sdt\right).\]
If \((f_\varepsilon)_\varepsilon\in \lbrace L^2([0,T];H^s(\mathbb{R}^n))\rbrace^{(0,1]}\) \[\begin{align} &\sup_{0\leq t \leq T}\|u_\varepsilon(t,\cdot)\|^2_s+\omega(\varepsilon)^{-k}\int_0^T \|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u_\varepsilon(t,\cdot)\|_0^2\,dt\\ & \leq Ce^{C_1\omega(\varepsilon)^{-k_1}T}\left(\|u_{0,\varepsilon}\|_s^2+\int_0^T \|f_\varepsilon(t,\cdot)\|_s^2 dt\right). \end{align}\]
If \((\Lambda^{s-\frac{m-1}{2}}f_\varepsilon)_\varepsilon\in \lbrace L^2([0,T] \times \mathbb{R}^n, \langle x \rangle^{-N}dxdt)\rbrace^{(0,1]}\) \[\begin{align} &\sup_{0\leq t \leq T}\|u_\varepsilon(t,\cdot)\|^2_s+\omega(\varepsilon)^{-k} \int_0^T \|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u_\varepsilon(t,\cdot)\|_0^2\,dt\\ &\leq Ce^{C_1\omega(\varepsilon)^{-k_1}T}\left(\|u_{0,\varepsilon}\|_s^2+\int_0^T \| \langle x \rangle^{N/2}\Lambda^{s-1/2}f_\varepsilon(t,\cdot)\|^2_0 \,dt\right), \end{align}\]
where \(C,C_1>0\) and \(k\), \(k_1 \in \mathbb{N}\), depend on the coefficient of \(P\) defined in 22 and on \(s\), but are independent of \(\varepsilon\).
Remark 26. Let us emphasise that, in the case of \(u_0 \in H^s(\mathbb{R}^n)\) and \(f\) having suitable regularity (as required in \((i)\), \((ii)\) or \((iii)\) of the previous remark), we have that the net of solutions \((u_\varepsilon)_\varepsilon\) satisfies the estimates \((i)\), \((ii)\) and \((iii)\) with \(u_0\) and \(f\) appearing on the right-hand sides instead of their regularised versions \((f_\varepsilon)_\varepsilon\) and \((u_{0,\varepsilon})_\varepsilon\) (cf. Theorem 23). It therefore follows that in this case, without regularising right-hand side and initial data, we can generate a very weak solution of \(H^s\)- type rather than \(H^\infty\).
We now prove a uniqueness result of an \(H^\infty\)-weak solution in the following sense (cf. [15]): if we perturb the coefficients of the regularised operator by suitable negligible nets, then the net of solution of the Cauchy problem associated with the perturbed operator will differ from \((u_\varepsilon)_\varepsilon\) by an \(H^\infty\)-negligible net (see Definition 10 below).
For this purpose, we perturb the coefficients and define the net of operators \((P'_\varepsilon)_\varepsilon\) by \[\label{eq46Pepsprimo} \begin{align} P_\varepsilon'&=D_t-A'_\varepsilon-B'_\varepsilon-V'_\varepsilon\\ &:=\sum_{i,j=1}^nD_{x_i}\bigl((a_{ij,\varepsilon}(x)+n_{ij,\varepsilon}(x))D_{x_j}\bigr) -\sum_{k=1}^n(b_{k,\varepsilon}(x)+n_k(x))D_{x_k}-(V_\varepsilon(x)+n_{V,\varepsilon}(x)), \end{align}\tag{38}\] where
The matrix \(\mathcal{N}_\varepsilon(x)=(n_{ij,\varepsilon}(x))_{i,j}\) is real and symmetric for all \(x \in \mathbb{R}^n\). Furthermore, \(n_{ij,\varepsilon} \in C^\infty(\mathbb{R}^n)\) and for all \(q \in \mathbb{N}_0\) and all \(\beta \in \mathbb{N}_0^n\) there exists \(C>0\) such that \[\sup_{x \in \mathbb{R}^n}|\partial_x^{\beta}n_{ij,\varepsilon}(x)|\leq C\varepsilon^{q}, \quad \forall i,j=1,\dots,n,\] for all \(\varepsilon\in (0,1]\). Finally, for each fixed \(i,j,k=1,\dots,n\), we assume \[|\partial_{x_k} n_{ij,\varepsilon}(x)| \leq \nu \langle x \rangle^{-N}, \quad \forall x \in \mathbb{R}^n, \;\forall \varepsilon\in (0,1],\] where \(\nu>0\) and \(N>1\) are the constants appearing in 3.
For each \(\beta \in \mathbb{N}_0^n\) and for each \(q \in \mathbb{N}_0\) there exists a constant \(C'>0\) such that for any \(k=1,\dots,n\) \[\sup_{x \in \mathbb{R}^n}|\partial_x^\beta n_{k,\varepsilon}(x)|\leq C'\varepsilon^{q}, \quad \forall \varepsilon\in (0,1].\] Moreover, there exists a universal constant \(c_0'>0\) such that, for each \(\varepsilon \in (0,1]\) (with \(N>1\) as in 3), \[|\mathrm{Im}(n_{k,\varepsilon})(x)| \leq c_0' \langle x \rangle^{-N}, \quad \forall x \in \mathbb{R}^n, \;\forall k = 1, \dots, n.\]
For each \(\beta \in \mathbb{N}_0^n\) and each \(q \in \mathbb{N}_0\), there exists a constant \(C''>0\) such that \[\sup_{x \in \mathbb{R}^n}|\partial_x^\beta n_{V,\varepsilon}(x)| \leq C\varepsilon^q, \quad \forall \varepsilon\in (0,1].\]
Furthermore, we consider an \(H^\infty(\mathbb{R}^n)\)-negligible perturbation of the Cauchy data in the sense of Definition 10. Specifically, we consider the Cauchy problem \[\label{eq46cauchyPprimo} \begin{cases} P_\varepsilon' u_\varepsilon=f_\varepsilon+n_{f,\varepsilon} \quad \text{in} \;(0,T] \times \mathbb{R}^n, \\ u_\varepsilon(0,\cdot)=u_{0,\varepsilon}+n_{u_0,\varepsilon} \quad \text{in}\; \mathbb{R}^n, \end{cases}\tag{39}\] where \(P_\varepsilon'\) is defined as in 38 , \((f_\varepsilon)\) and \((u_{0,\varepsilon})_\varepsilon\) are \(H^\infty\)-moderate nets and \((n_{f,\varepsilon})_\varepsilon\), \((n_{u_0,\varepsilon})_\varepsilon\) are \(H^\infty\)-negligible nets.
To ensure the existence of a solution of the problem 39 , we need the following proposition.
Proposition 27. Let \((P_\varepsilon')_\varepsilon\) be the net of operators defined as in 38 . Then, 1, 2, 3, 4, 5 hold for \((P_\varepsilon')_\varepsilon\), provided \(\varepsilon\in (0,\varepsilon_0]\), with \(\varepsilon_0>0\) sufficiently small.
Proof. In this framework condition 1, 3, 4 and 5 are trivially satisfied. Therefore, we only need to check 2. To do that we note that (recall \(\mathcal{A}_\varepsilon(x)=(a_{ij,\varepsilon}(x))_{i,j}\) and \(\mathcal{N}_\varepsilon(x)=(n_{ij,\varepsilon})_{i,j})\)) \[|(\mathcal{A}_\varepsilon(x)+\mathcal{N}_\varepsilon(x))\cdot \xi | \geq |\mathcal{A}_\varepsilon(x) \cdot \xi|-|\mathcal{N}_\varepsilon(x) \cdot \xi| \geq \mu |\xi|-c'\varepsilon^q|\xi| \geq c |\xi|, \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n,\] for some \(c',c>0\), independent of \(\varepsilon\), provided \(\varepsilon\in (0,\varepsilon_0]\). ◻
Thus, since \((P_\varepsilon')_\varepsilon\) satisfies the aforementioned conditions there exists \((u_\varepsilon')_\varepsilon\) \(H^\infty\)-weak solution of 39 . Then, for each \(\varepsilon\in (0,\varepsilon_0]\), with \(\varepsilon_0\) given as in Proposition 27, we have \[\begin{cases} &P_\varepsilon(u_\varepsilon-u_\varepsilon')=-n_{f,\varepsilon}-(P_\varepsilon-P_\varepsilon')u_\varepsilon'\\ &u_\varepsilon(0,\cdot)-u_\varepsilon'(0,\cdot)=-n_{u_0,\varepsilon}. \end{cases}\] Therefore, by ?? we have that, for each \(s \in \mathbb{R}\), there exist \(C>0\) and \(N \in \mathbb{N}\) so that \[\sup_{0\leq t\leq T}\|u_\varepsilon(t,\cdot)-u_\varepsilon'(t,\cdot)\|_s\leq C\varepsilon^{-N}\left(\|n_{u_0,\varepsilon}\|_s+\int_0^T \|n_{f,\varepsilon}(t,\cdot)+(P_\varepsilon-P_\varepsilon')u_\varepsilon'(t,\cdot)\|_sdt\right),\] for all \(\varepsilon\in (0,\varepsilon_0]\). Hence, since \((n_{u_0,\varepsilon})_\varepsilon\) and \((n_{f,\varepsilon})_\varepsilon\) are \(H^\infty\)-negligible, and the net of coefficients of \((P_\varepsilon-P_\varepsilon')_\varepsilon\) satisfies negligible estimates (cf. [15]) we may conclude that for all \(q>0\) there exists \(c>0\) such that \[\sup_{0\leq t\leq T}\|u_\varepsilon(t,\cdot)-u_\varepsilon'(t,\cdot)\|_s\leq c\varepsilon^{q}, \quad \forall \varepsilon\in (0,\varepsilon_0].\] This proves the uniqueness result we wanted to obtain.
startsection section1@-3.5ex plus -1ex minus -.2ex2.3ex plus .2exConsistency with the classical theory In this section, we establish consistency with the classical theory. This means that when the coefficients are smooth, we recover the classical solution as \(\varepsilon\rightarrow 0\). We will consider first a problem with diagonal principal part and then we will deal with the general set-up.
We begin by recalling the classical result proven in [4] and [5] for evolution equation of the form \[\begin{cases}\label{mainprobclass} Pu=f \quad \text{in} \;]0,T] \times \mathbb{R}^n, \\ u(0,\cdot)=u_0 \quad \text{in} \;\mathbb{R}^n, \end{cases}\tag{40}\] where \(u_0 \in H^s(\mathbb{R}^n)\), for \(s \in \mathbb{R}\), f has suitable regularity (see Theorem 28 below) and \[\label{eq46Pclass} P=D_t-\sum_{i,j=1}^nD_{x_i}(a_{ij}(x)D_{x_j})- \sum_{k=1}^nb_k(x)D_{x_k}-V(x),\tag{41}\] with \(a_{ij},b_k,V \in C_b^\infty(\mathbb{R}^n)\) that satisfy the following hypotheses:
The matrix of the coefficients \(\mathcal{A}(x)\) is real and symmetric for each \(x \in \mathbb{R}^n\), and satisfies the non degeneracy condition: there exists \(\mu>0\) such that \[\label{eq46condclass1} \mu^{-1}|\xi| \leq |\mathcal{A}(x) \cdot \xi| \leq \mu |\xi|, \quad \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n.\tag{42}\]
There exist \(\nu>0\) sufficiently small and \(N \in \mathbb{N}\), \(N>1\), such that for all \(\alpha \in \mathbb{N}_0^n\), with \(|\alpha|\leq 1\), one has \[\label{eq46condclass2} |\partial_{x}^{\alpha} a_{ij}(x)| \leq \nu \langle x \rangle^{-N}, \quad \forall x \in \mathbb{R}^n, \;\;\forall i,j=1,\dots,n.\tag{43}\]
There exists a universal constant \(c_0>0\) such that, \[\label{eq46condclass3} |\mathrm{Im}(b_k)(x)| \leq c_0 \langle x \rangle^{-N}, \quad \forall x \in \mathbb{R}^n, \;\forall k = 1, \dots, n,\tag{44}\] with \(N\in \mathbb{N}\) as above, and for each \(\beta \in \mathbb{N}_0^n\) \[\label{eq46condclass4} |\partial_x^\beta b_k(x)|\leq C_\beta, \quad \forall x \in \mathbb{R}^n, \;\forall k = 1, \dots, n,\tag{45}\] for some constant \(C_\beta>0\).
For each \(\gamma \in \mathbb{N}_0^n\) \[\label{eq46condclass5} |\partial_x^\gamma V(x)| \leq C_\gamma, \quad \forall x \in \mathbb{R}^n,\tag{46}\] for some constant \(C_\gamma>0\)
Theorem 28. Let \(u_0 \in H^s(\mathbb{R}^n)\), and let \(N>1\) as above. Then, under the aforementioned hypotheses on \(P\), we have the following results.
If \(f\in L^1([0,T];H^s(\mathbb{R}^n))\), the IVP 40 has a unique solution \(u \in C([0,T];H^s(\mathbb{R}^n))\) satisfying
\[\label{eq46iclass} \sup_{0\leq t\leq T}\|u(t,\cdot)\|_s\leq C_2e^{TC_1}\left(\|u_0\|_s+\int_0^T \|f(t,\cdot)\|_sdt\right).\qquad{(19)}\]
If \(f\in L^2([0,T];H^s(\mathbb{R}^n))\), the IVP 40 has a unique solution \(u\in C([0,T];H^{s}(\mathbb{R}^n))\) satisfying \[\label{eq46iiclass} \begin{align} &\sup_{0\leq t \leq T}\|u(t,\cdot)\|^2_s+\int_0^T \|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u(t,\cdot)\|_0^2\,dt\\ & \leq C_2e^{TC_1}\left(\|u_0\|_s^2+\int_0^T \|f(t,\cdot)\|_s^2 dt\right). \end{align}\qquad{(20)}\]
If \(\Lambda^{s-\frac{m-1}{2}}f\in L^2([0,T]\times \mathbb{R}^n;\langle x \rangle^N dx dt)\), the IVP 40 has a unique solution \(u\in C([0,T];H^{s}(\mathbb{R}^n))\) satisfying \[\label{eq46iiiclass} \begin{align} &\sup_{0\leq t \leq T}\|u(t,\cdot)\|^2_s+ \int_0^T \|\langle x \rangle^{-{N/2}}\Lambda^{s+1/2}u(t,\cdot)\|_0^2\,dt\\ &\leq C_2e^{TC_1}\left(\|u_0\|_s^2+\int_0^T \| \langle x \rangle^{N/2}\Lambda^{s-1/2}f(t,\cdot)\|^2_0 \,dt\right), \end{align}\qquad{(21)}\]
for some constants \(C_1,C_2>0\) depending on \(s\) and on the coefficients of \(P\).
Remark 29. The proof of this theorem is the same as in [5] and [4]. The result follows from a priori estimates, established, for instance, in Lemma 2.3.1 of [5], together with a standard functional analysis argument. The key tool for obtaining such estimates is the so-called Doi’s lemma (see, for instance, Lemma 2.2.2 in [5]), which in [5] is derived from a non-trapping condition on the bicharacteristic curves of the principal symbol of \(A\). In the present setting, the non-trapping condition is replaced by the smallness condition 43 , under which Doi’s lemma still holds (cf. [4]). Hence, the proof of the a priori estimates, which (once again together with a standard functional analysis argument) leads to the proof of Theorem 28, is the same as that in Lemma 2.3.1 of [5].
The next goal is to prove that whenever \(P\) is an operator with smooth coefficients satisfying suitable conditions (which, in turn, imply conditions 42 –46 and therefore guarantee well-posedness and the smoothing effect in the classical sense), the corresponding net of regularised operators\((P_\varepsilon)_\varepsilon\) satisfies 1–5. By repeating the proof of Theorem 20 we get the following Proposition.
Proposition 30. Let \(P\) be an operator of the form \[P=D_t-\sum_{i,j=1}^nD_{x_i}(a_{ij}(x)D_{x_j})- \sum_{k=1}^nb_k(x)D_{x_k}-V(x),\] and assume that the coefficients fulfill the following hypotheses.
The matrix \(\mathcal{A}(x)=(a_{ij}(x))_{i,j=1}^n\) is real and symmetric, for all \(x \in \mathbb{R}^n\) and its entries can be decomposed as \[a_{ij}(x)=c_{ij}+\tilde{a}_{ij}(x), \quad i,j=1,\dots,n,\] where \(c_{ij} \in \mathbb{R}\) and \(\tilde{a}_{ij} \in C_b^\infty(\mathbb{R}^n)\), for all \(i,j=1,\dots,n\). Moreover, we assume that the matrix \(\mathcal{C}=(c_{ij})_{i,j=1}^n\) is non-degenerate, namely there exists \(\mu'>0\) such that \[{\mu'}^{-1}|\xi|\leq |\mathcal{C} \cdot \xi |\leq \mu' |\xi|, \quad \forall \xi \in \mathbb{R}^n,\] and for all \(i,j=1,\dots,n\) the perturbations satisfy \[|\partial_x^{\alpha}\tilde{a}_{ij}(x)|\leq \nu' \langle x \rangle^{-N},\quad \forall |\alpha|\leq 1, \;\forall x \in \mathbb{R}^n,\] where \(N \in \mathbb{N}\), \(N>1\) and \(\nu'>0\) is sufficiently small.
For all \(k=1,\dots,n\), we assume \(b_k \in C^\infty(\mathbb{R}^n)\) and (with \(N>1\) as above) \[|\mathrm{Im}(b_k)(x)|\leq c_0'\langle x \rangle^{-N},\] for some universal constant \(c_0'>0\).
\(V \in C^\infty_b(\mathbb{R}^n)\).
Then, the corresponding regularised operators \((P_\varepsilon)_\varepsilon\) defined as in Section [sec:sec46stateandreg], fulfills 1-5.
Remark 31. Note that if \(P\) is an operator that satisfies the hypotheses of Proposition 30, then it satisfies 42 46 .
Remark 32. It is remarkable that the consistency result can also be applied to a simpler class of (regular) operators tailored to Example 15. More precisely, let \(P\) be the operator defined in 41 with diagonal matrix \(\mathcal{A}\), i.e. \[P=D_t-\sum_{i=1}^nD_{x_i}(a_{ii}(x)D_{x_i})- \sum_{k=1}^nb_k(x)D_{x_k}-V(x).\] We assume that \[a_{ii}(x)=c_{i}+\tilde{a}_{ii}(x),\] where \(c_{ii}\in \mathbb{R} \setminus \lbrace 0\rbrace\) is a constant and \(\tilde{a}_{ii}\) is smooth and bounded for \(i=1,\cdots, n\) with \[0\le \tilde{a}_{ii}(x)\le \frac{|c_i|}{2},\] for all \(x\). Furthermore, we require that the lower order parts of \(P\) satisfy the same hypotheses \((ii)\) and \((iii)\) of Proposition 30.
Let \((P_\varepsilon)_\varepsilon\) be the corresponding regularised operators as in Section [sec:sec46stateandreg]. If \(P\) fulfills the hypotheses 42 46 then \((P_\varepsilon)_\varepsilon\) fulfills the hypotheses \((H1)-(H5)\).
The proof of this fact follows by repeating the proof of Theorem 20 in the case where \(P\) is an operator with regular coefficients.
We are now ready to prove the following consistency result.
Let \(P\) be an operator satisfying the hypotheses of Proposition 30 and let \(f \in C([0,T];H^\infty(\mathbb{R}^n))\) and \(u_0 \in H^\infty(\mathbb{R}^n)\). Let also \((f_\varepsilon)_\varepsilon\) and \((u_{0,\varepsilon})_\varepsilon\), with \(f_\varepsilon(t,x)=f_\varepsilon\ast \varphi_\varepsilon\) and \(u_{0,\varepsilon}=u_0 \ast\varphi_\varepsilon\), for \(\varepsilon\in (0,1]\) and \(\varphi \in \mathscr{S}(\mathbb{R}^n)\), \(\int \varphi \, dx=1\), with all vanishing moments, i.e. \(\int x^\alpha \varphi(x)dx=0\), \(\forall \alpha \neq 0\).
We denote by \(u\) the solution of \[\begin{cases} P u=f \quad \text{in} \;\;(0,T] \times \mathbb{R}^n \\ u(0,\cdot)=u_0\quad \text{in}\;\;\mathbb{R}^n, \end{cases}\] that we recall it exists and is unique due to Theorem 28 and Remark 29, and by \(u_\varepsilon\) the solution of \[\begin{cases} P_\varepsilon u_\varepsilon=f_\varepsilon\quad \text{in} \;\;(0,T] \times \mathbb{R}^n \\ u_\varepsilon(0,\cdot)=u_{0,\varepsilon}\quad \text{in}\;\;\mathbb{R}^n. \end{cases}\]
Proposition 33. Under the hypotheses above any very weak solution \(u_\varepsilon\) converges to the classical solution \(u\) in \(C([0,T];H^\infty(\mathbb{R}^n))\).
Proof. Note that \(u-u_\varepsilon\) solves, \[\begin{cases} P(u-u_\varepsilon)=(f-f_\varepsilon)+R_\varepsilon u_\varepsilon\quad \text{in} \;\;(0,T] \times \mathbb{R}^n \\ (u-u_\varepsilon)(0,\cdot)=u_0-u_{0,\varepsilon}\quad \text{in}\;\;\mathbb{R}^n, \end{cases}\] with \[\label{eq46reps} R_\varepsilon=P_\varepsilon-P=\sum_{i,j=1}^nD_{x_i}\bigl((a_{ij,\varepsilon}(x)-a_{ij}(x))D_{x_j}\bigr)- \sum_{k=1}^n(b_{k,\varepsilon}(x)-b_k(x))D_{x_k}-(V_\varepsilon(x)-V(x)).\tag{47}\] Therefore, denoting \(R_\varepsilon=\mathrm{Op}(r_\varepsilon)\), by Theorem 1, Theorem 28 and Remark 31 we have \[\begin{align} \sup_{0\leq t\leq T}\|u(t,\cdot)-u_\varepsilon(t,\cdot)\|_s & \leq C\Bigl(\|u_0-u_{0,\varepsilon}\|_s+\int_0^T \|(f-f_\varepsilon)(t,\cdot)\|_sdt+ \int_0^T \|R_\varepsilon u_\varepsilon(t,\cdot)\|_sdt \Bigr)\\ & \leq C'\Bigl(\|u_0-u_{0,\varepsilon}\|_s+\int_0^T \|(f-f_\varepsilon)(t,\cdot)\|_sdt+ |r_\varepsilon|^{(2)}_k\int_0^T \| u_\varepsilon(t,\cdot)\|_{s+2}dt \Bigr), \end{align}\] for some \(C,C',k\) independent of \(\varepsilon\). We now note that, by reasoning as in [15] we have that, since the coefficients of the operator \(P\) are \(C^\infty\), we obtain estimate ?? uniformly with respect to \(\varepsilon\), that lead to \[\|u_\varepsilon(t,\cdot)\|_{s+2}\leq C_s,\] for some \(C_s>0\), uniformly in \(\varepsilon\). Moreover, since \(R_\varepsilon=\mathrm{Op}(r_\varepsilon)\) is given by 47 , we easily obtain that \[|r_\varepsilon|_k^{(2)} \rightarrow 0, \quad \text{as} \;\varepsilon\rightarrow 0^+.\] Finally, \((f-f_\varepsilon)\varepsilon\) and \((u_0-u_{0,\varepsilon})_\varepsilon\) are \(H^\infty\)-negligible nets, since the Cauchy data \(u_0\) and \(f\) have been regularised by means of a mollifier with all vanishing moments (cf. [15]).
Therefore, in conclusion \[u_\varepsilon\rightarrow u \quad \text{as} \;\varepsilon\rightarrow 0, \quad \text{in} \;\;C([0,T];H^\infty(\mathbb{R}^n)),\] and this proves the consistency of our approach with the classical theory. ◻
Remark 34. We conclude this work by emphasising that our approach allows us to obtain a convergence result as \(\varepsilon \to 0\) even when the coefficients of the operator \(P\), generating problem 1 , are merely \(C^k\) in the spatial variables (rather than \(C^\infty\), as required in the classical theory), for a sufficiently large integer \(k\). The Cauchy data \(f\) and \(u_0\) are assumed to belong to the Sobolev space \(H^s(\mathbb{R}^n)\) for some \(s \in \mathbb{R}\).
More precisely, consider the Cauchy problem \[\begin{cases} Pu = f & \text{in } (0,T] \times \mathbb{R}^n, \\ u(0,\cdot) = u_0 & \text{in } \mathbb{R}^n, \end{cases}\] where \(u_0 \in H^s(\mathbb{R}^n)\), \(f \in C([0,T]; H^s(\mathbb{R}^n))\), and \(P\) is defined as in 21 , with coefficients in \(C^k(\mathbb{R}^n)\) for some \(k\), satisfying the structural conditions stated above.
By repeating the regularisation argument used in the proof of Theorem 23, we may construct a net of solutions \((u_\varepsilon)_\varepsilon\) solving the corresponding regularised problems induced by the net \((P_\varepsilon)_\varepsilon\), with the same initial data \(u_0\) and right-hand side \(f\). Moreover, if the coefficients of \(P\) are sufficiently regular (i.e. is \(k\) is sufficiently large) all the seminorms of symbols that appear in the proof of Theorem 23 actually do not depend on \(\varepsilon\). Therefore, by repeating the proof, we obtain that \[\sup_{0 \le t \le T} \|u_\varepsilon(t,\cdot)\|_{H^s} \le C_2 e^{C_1 T} \left( \|u_0\|_{H^s} + \int_0^T \|f(t,\cdot)\|_{H^s}\, dt \right),\] uniformly in \(\varepsilon\). Hence, the net \((u_\varepsilon(t,\cdot))_\varepsilon\) is uniformly bounded in \(\varepsilon\) in \(H^s(\mathbb{R}^n)\), for each \(t \in [0,T]\) and then it admits a subsequence converging weakly in this space.