January 01, 1970
In this paper, we investigate the orbital stability of vortex patches for the two-dimensional incompressible Euler equations in both a class of domains that satisfy the “weak finite volume condition” and a strip of arbitrary width. We establish that for suitable parameters \((\mu,\lambda)\), the penalized kinetic energy functional admits a minimizer, and that every such minimizer satisfies the elliptic equation \(\omega = \lambda(\psi - W x_2 - \gamma)_+\). Furthermore, we demonstrate that the set of minimizers is orbitally stable under the Eulerian dynamics. This work extends the variational framework developed by Abe and Choi [1] to domains that lack both spatial scaling invariance and horizontal translation invariance. The absence of these properties introduces substantial difficulties in the proof, as classical rearrangement and scaling arguments are no longer applicable. We overcome these obstacles by comparing the Green’s function with that of the half-plane and exploiting the decay condition to formulate a concentration-compactness argument that ultimately yields the desired stability result.
The two-dimensional Euler equation for the motion of an inviscid, incompressible fluid on a fixed domain \(\Omega\subseteq \mathbb{R}^2_+\) is given in vorticity form by \[\begin{align} \partial_t \omega +u\cdot\nabla \omega&= 0, ~ \quad \omega(x,0)=\omega_0(x) \quad \text{ in } \Omega. \label{eqn:vorticity32equation} \end{align}\tag{1}\] Here, \(\omega\) is the vorticity of the flow, and the fluid velocity \(u\) satisfies the slip boundary condition: \[\begin{align} u\cdot \mathcal{N}=0, \end{align}\] where \(\mathcal{N}\) is the unit outward normal. When the appropriate Biot-Savart law is applicable, \(u\) is determined from \(\omega\) in the sense that \[u(x,t)=\nabla^\perp \int_D G(x,y)\omega(y,t)dy,\] and \(G\) is the Green’s function for the Dirichlet problem in \(\Omega\). It will be convenient for us to define the stream function associated with \(\omega\): \[\begin{align} \psi(x,t)=\int_{\Omega} G(x,y) \omega(y,t)dy, \label{def:stream32function} \end{align}\tag{2}\] so that \(u(x,t)=\nabla^\perp \psi\). In addition, during this paper, when conducting a discussion at a fixed time, the time \(t\) will be omitted, i.e., \(\omega (\cdot, t) = \omega (\cdot), \psi (\cdot, t) = \psi (\cdot)\) and \(u(\cdot, t) = u(\cdot)\).
A natural question is whether solutions remain stable over long time intervals. In general, the answer is negative. For 2D Euler equations in a disk, Nadirashvili, Alexander, and Šverák [2] constructed solutions whose vorticity gradient grows double‑exponentially. Similar mechanisms were exploited on the torus \(\mathbb{T}^2\) by Zlatoš, Andrej [3], applied to smooth domains with an axis of symmetry by Xu [4], and also for free-boundary problems by Hu, Luo, and Yao [5], demonstrating that instability can occur generically.
Nevertheless, stability can be established for certain domains with suitable classes of initial data. Cao and Wang [6] proved nonlinear stability of highly concentrated vortex patches near non‑degenerate minima of the Robin function; Cao, Wan, and Wang [7] showed orbital stability for kinetic‑energy‑maximising patches; Choi, Jeong, and Lim [8] studied the stability of monotone, nonnegative, and compactly supported vorticities in the half cylinder. Earlier work by Burton [9] and Turkington [10] established foundational stability results for vortex patches using rearrangement techniques. However, all these results highly rely on the specific geometry of the domain, the assumptions about the initial data, and the techniques used (such as rearrangement or translation invariance).
In recent years, a variational approach based on minimizing a penalized energy has been successfully applied to prove the orbital stability of vortex patches. Abe and Choi [1] established the stability of Lamb dipoles in the half‑plane \(\mathbb{R}_+^2\) by considering the minimization problem \[\begin{align} I_{\mu,\nu,\lambda}=\inf_{\omega \in K_{\mu,\nu}}\{-E_{2,\lambda}\}, \label{def:minimizing32problem} \end{align}\tag{3}\] where \(K_{\mu,\nu}\) is a class of admissible vorticities with fixed first moment and bounded \(L^1\) norm, and \(E_{2,\lambda}\) is the penalized kinetic energy. Their proof relies on two key properties of the half‑plane: i) the translation invariance property, which allows the use of rearrangement techniques, and ii) the spatial scaling invariance, which is used to obtain monotonicity and negativity and strictly decreasing in \(\mu\) of \(I_{\mu,\nu,\lambda}\) to control minimizing sequences.
The variational approach of Abe and Choi [1] has recently been extended to more complex symmetric configurations. Choi, Jeong, and Yao [11] studied the orbital stability of a pair of opposite-signed Lamb dipoles under odd-odd symmetry, as well as concentrated vortices in a quadrant, using the monotonicity of the first moment and pointwise kernel estimates. Abe, Choi, Jeong, Sim, and Woo [12] demonstrate the existence and stability of Sadovskii vortices. However, those results are heavily reliant on the convenience of the (half) plane, i.e., the explicit structure of the Green’s function, the scaling property, and rearrangement techniques. In contrast, domains meet the “weak finite volume condition” and infinite strip considered here lack these structures, requiring new tools to establish weak continuity of the kinetic energy and consequently the existence and stability of minimizers.
The main goal of this paper is to extend the variational approach to two classes of domains:
\(\Omega=\mathcal{D}= D_u\cup D_l\), where \[\begin{align} D_u := \mathcal{D} \cap \{x_2\ge 1\}, && D_l := \mathcal{D}\cap \{x_2<1\}, \label{def:upper32and32lower32part32of32domain} \end{align}\tag{4}\] that
satisfies the following two conditions:
The “weak finite volume condition”, i.e., \[\begin{align} \int_{D_l} x_2^p dx<\infty, \text{ for some } p\geq 0,\text{ and } ~ \text{Vol}(D_u) <\infty.\label{def:weak95finite95volume95domain95assumption} \end{align}\tag{5}\]
The Biot-Savart law is applicable on \(\mathcal{D}\).
Without loss of generality, it suffices to study the case when \(p\ge 2\). This class includes a wide range of domains, including but not limited to all simply connected domains with (piecewise) smooth boundary [13]. The weak tangency condition on non-smooth domains is discussed in [14] and [15]. In particular, this class covers all bounded domains, domains with periodic boundaries, and domains whose boundaries are graphs of some appropriate functions. These are of considerable importance in the study of free-boundary problems.
\(\Omega=\mathcal{S}\), where \(\mathcal{S}\) is the infinite strip \[\begin{align} \mathcal{S}=\mathbb{R}\times (0,L), ~ L>0, \label{def:strip95domain95assumption} \end{align}\tag{6}\] whose Green’s function is given explicitly by \[\begin{align} G(x,y)=&\frac{1}{4\pi}\ln\Big(1+\frac{\sin(\frac{\pi}{L}x_2)\sin(\frac{\pi}{L}y_2)}{\sinh(\frac{\pi}{2L}(x_1-y_1))^2+\sin(\frac{\pi}{2L}(x_2-y_2))^2}\Big). \label{def:green39s32function32for32strip} \end{align}\tag{7}\] Such strips can be regarded as a generalization of the half-plane model to domains of finite width, or as a limit \(p\to \infty\) of Class [def:case32Omega61D]. Nevertheless, due to the different widths \(L\) and the absence of decay of the domain itself prevent a direct application of the methods from [1] or from Class [def:case32Omega61D].
More precisely, we introduce the set of admissible vorticities for parameters \(0\leq \mu,\nu,\lambda < \infty\): \[\begin{align} K_{\mu,\nu}=\left\{\omega\in L^2(\Omega) \big| \omega\geq0, \int_{\Omega}x_2\omega(x)dx=\mu, \int_{\Omega}\omega(x)dx\leq\nu\right\}. \end{align}\]
The kinetic energy and the penalized energy are given by \[\begin{align} E[\omega]=\frac{1}{2}\int_{\Omega}\int_{\Omega}G(x,y)\omega(x)\omega(y)dxdy,\quad E_{2,\lambda}[\omega]=E[\omega]-\frac{1}{2\lambda}\|\omega\|_{2}^2, \end{align}\] where \(G\) is the Green’s function for the domain \(\Omega\). Define \(S_{\mu,\nu,\lambda}\subseteq K_{\mu,\nu}\) by the set of minimizers for 3 in \(K_{\mu,\nu}\). By the measure scaling, \(\hat{\omega}(x)=\omega(x)/\nu,\) the problem reduces to the case \(\nu=1\). We abbreviate the notation as \[\begin{align} K_\mu=K_{\mu,1}, ~ I_{\mu,\lambda}=I_{\mu,1,\lambda}, \text{ and } S_{\mu,\lambda}=S_{\mu,1,\lambda}. \end{align}\] Recall the the stream function associated with vorticity \(\omega\) in 2 , \(E[\omega]=\frac{1}{2}\int \psi\omega dx\). Adapting the proof of [1] to our setting encounters three main difficulties that demand new analytical tools.
Due to the loss of spatial scaling invariance, the Green’s function on \(\Omega\) does not satisfy the property that \(G(cx,cy)=G(x,y)\). Therefore, for any \(c>0\) and scaling \(\widehat\omega (x)=\omega(cx)\), there is no possible connection between \(E[\widehat\omega|_{\Omega}]\) and \(E[\omega]\). Meanwhile, if we consider the synchronous changes of the domain \(\Omega\) during scaling and regard \(\widehat\omega\) as a function defined on \(c^{-1}\Omega\), the penalized energy of \(\widehat\omega\) should be: \[\begin{align} E^c_{2,\lambda}[\widehat\omega] = & \int_{c^{-1}\Omega}\int_{c^{-1}\Omega}G^{c}(x,y)\widehat\omega(x)\widehat\omega(y)dxdy- \frac{1}{2\lambda}\int_{c^{-1}\Omega}\widehat\omega^2dx\\ = & c^{-4}\int_{\Omega}\int_{\Omega}G(x,y)\omega(x)\omega(y)dxdy-\frac{1}{2\lambda}\int_{c^{-1}\Omega}\widehat\omega^2dx\\ = & c^{-4}E[\omega] - c^{-2}\frac{1}{2\lambda}\int_{\Omega}\omega^2dx, \end{align}\] where we temporarily use \(E^{c}\) and \(G^c\) to represent the kinetic energy and the Green’s function for \(c^{-1}\Omega=\left\{c^{-1}x:x\in \Omega \right\}\). To compare \(E_{2,\lambda}^{c}[\widehat\omega]\) with \(E_{2,\lambda}[\omega]\), it is necessary to let \(c^{-4}=c^{-2}\) with an unique solution \(c=1\), which is meaningless.
This issue directly led to the failure of the methods used in [1] to study kinetic energy and penalized energy. Fortunately, based on the inclusion relationship \(\Omega\subseteq \mathbb{R}^2_+\), we have \[\begin{align} 0\leq G(x,y)\leq G_H(x,y) \text{ for any } x,y\in \overline{\Omega},\label{ineq:estimate32of32green39s32function} \end{align}\tag{8}\] where \(G_H\) is the Green’s function for the upper half plane, and this comparison allows us to derive estimates similar to those in [1]. By choosing appropriate parameters \(\mu,\lambda\) and constructing explicit trial functions in \(K_{\mu,\lambda}\), we prove negativity of \(I_{\mu,\nu,\lambda}\). Other impacts of the loss of scaling will be mentioned and solved below.
The Green’s function of domains in this class cannot be written as \(G(x,y)=g(|x_1-y_1|,x_2,y_2)\) for some function \(g\), so that the kinetic energy is not translation invariant. Meanwhile, such difficulty also undermines all techniques that rely on rearrangement methods.
To overcome this, we introduce the weak finite volume condition of \(\mathcal{D}\) itself to show that the weighted mass of any minimizing sequence concentrates on a bounded region. In particular, for any minimizing sequence \(\{\omega_n\}\) and \(\epsilon>0\), there exists a bounded domain \(D_0\subseteq \mathcal{D}\) such that the kinetic energy, as well as crossing terms, involving \(\omega_n|_{\mathcal{D}\setminus D_0}\) is in \(\mathcal{O}(\epsilon)\). Under such a split, the Green’s function is square-integrable on \(D_0\times D_0\), so that we can pass the limit in kinetic energy using weak \(L^2\) convergence and obtain the existence of a minimizer and a general convergence theorem.
The method in [1], as well as [16] and [17], applies to prove compactness of a minimizing sequence satisfying \[\begin{align} \begin{aligned} & \omega(x_1,x_2)=\omega(-x_1,x_2),\\ & \omega(x_1,x_2) \text{ is non-increasing for } x_1 > 0. \end{aligned}\label{def:symmetry32property} \end{align}\tag{9}\] Under such an assumption, these authors discuss the weighted mass of the minimizing sequence of \(I_{\mu,\nu,\lambda}\) is concentrated under a proper \(x_1\)-translation, and the key to this lies in the rearrangement and the strictly decreasing nature of \(I_{\mu,\nu,\lambda}\). The former can be resolved by two points: i) the structure of the Green’s function for \(\mathcal{S}\) meet the criteria of standard rearrangement inequality (see Proposition 14), ii) the comparison 8 leads to a comparison between stream functions by viewing \(L^i(\mathcal{S})=\{\omega\in L^i(\mathbb{R}_+^2) ~ | ~ \operatorname{spt}\omega\subseteq \overline{\mathcal{S}} \}\), then for any proper \(i>1\) and \(\omega\in L^i(\mathcal{S})\) can generate a stream function \(\psi^H\) on \(\mathbb{R}^2_+\) in the sense that \[\begin{align} \psi^H(x)=\int_{\mathbb{R}^2_+}G_H(x,y)\omega(y)dy=\int_{\mathcal{S}}G_H(x,y)\omega(y)dy, \end{align}\] with \[\begin{align} \psi(x)\leq \psi^H(x) \text{ for any } x\in \mathcal{S}. \label{ineq:comparsion32between32stream32function} \end{align}\tag{10}\] The latter is more complicated due to the lack of scaling invariance. We cannot follow the same route, instead, it is necessary to proof it alone a new path: prove the existence of a minimizer (see Sec 4.1), and then demonstrate more properties of minimizers as \(\mu\) is sufficiently small, such as an uniform \(L^1\) bound, to establish the strict monotonicity of \(I_{\mu,\nu,\lambda}\) (Proposition 21). This strict monotonicity is finally employed in a concentration-compactness framework to obtain a general convergence theorem for minimizing sequences (Theorem 23).
The paper is organized as follows. Section 2 contains preliminary estimates, Proposition 1 and 2, establishing the negativity of \(I_{\mu,\lambda}\) for suitable parameters and the uniform \(L^2\) boundedness of minimizing sequences. Consequently, we investigate the general structure of the minimizer in Proposition 6.
Section 3 is devoted to a general convergence theorem to the variational problem 3 for the Class [def:case32Omega61D], including the discussion about the kinetic energy (Proposition 8), and the existence of minimizer from minimizing sequence (Theorem 9). Section 4 is devoted to a general convergence theorem for the Class [def:case32Omega61S], including the discussion about the existence of minimizer (Sec 4.1), the uniformly \(L^1\)-norm decay to minimizers (Proposition 18), and the general convergence theorem (Theorem 23).
In Section 5, we prove the orbital stability of the minimizer over the lifespan (Theorem 24); finally, in Section 6, we present a discussion of future work.
Both authors acknowledge support from Hong Kong RGC Grants CUHK–14304424 and CUHK–14301225.
We establish the basic properties of the minimization problem. Thanks to the observation 8 , we derive several useful bounds for the kinetic energy and crossing terms. These are collected in Proposition 1 and are analogous to those obtained in [1]. Next, we study the sign of \(I_{\mu,\lambda}\). The proof of negativity differs from that in [1] because we lack scaling invariance; instead, we construct an explicit trial function supported on a finite measure subset of \(\Omega\). See Proposition 2 below.
Proposition 1. The following estimates hold for \(\omega,\omega_i\in L^1\cap L^2(\Omega)\) satisfying \(x_2\omega, x_2\omega_i\in L^1(\Omega)\) with some constant \(C\) independent from \(\omega,\omega_i, i=1,2\): \[\begin{align} &\Bigg|\int_{\Omega}G(x,y)\omega(y)dy\Bigg|\leq C x_2^{1/2}\|\omega\|_{1}^{1/2}\|\omega\|_{2}^{1/2},\label{ineq:estimate32of32stream32fct}\\ &E[\omega]\leq C \|x_2\omega\|_{1}^{1/2}\|\omega\|_{1}\|\omega\|_{2}^{1/2},\label{ineq:estimate32of32kinetic32energy}\\ &\Bigg|\int_{\Omega}\int_{\Omega}G(x,y)\omega_1(y)\omega_2(x)dxdy\Bigg|\leq C \|\omega_1\|_{1}^{1/2}\|\omega_1\|_{2}^{1/2}\|x_2\omega_2\|_{1}^{1/2}\|\omega_2\|_{1}^{1/2},\label{ineq:estimate32of32energy32crossing32term}\\ &|E[\omega_1]-E[\omega_2]|\leq C \|\omega_1-\omega_2\|_{1}^{1/2}\|\omega_1-\omega_2\|_{2}^{1/2}\|x_2(\omega_1+\omega_2)\|_{1}^{1/2}\|\omega_1+\omega_2\|_{1}^{1/2}.\label{ineq:estimate32of32energy32difference} \end{align}\] {#eq: sublabel=eq:ineq:estimate32of32stream32fct,eq:ineq:estimate32of32kinetic32energy,eq:ineq:estimate32of32energy32crossing32term,eq:ineq:estimate32of32energy32difference}
Proof: Use the inequality \(0\leq G(x,y)\leq G_H(x,y)\), we derive the bound for any \(q\in(1,2)\) that \[\begin{align} \Bigg(\int_{\Omega}G(x,y)^qdy\bigg)^{1/q}\leq \Bigg(\int_{\mathbb{R}^2_+}G_H(x,y)^qdy\Bigg)^{1/q}\leq Cx_2^{2/q}. \end{align}\] Then all estimates ?? ?? can be obtained through the same arguments in Proposition 2.1, [1].
These estimates, while elementary, are fundamental for the rest of the paper. They not only provide uniform bounds for general minimizing sequences (as will be seen in Remark 4) but also serve as the workhorse for controlling error terms in convergence arguments, such as the proof of Proposition 8 and 16.
Next, we shall prove that \(I_{\mu,\lambda}\) is negative for a suitable class of parameters \((\mu,\lambda)\). Different from Lemma 2.3 in [1], where \(I_{\mu,\lambda}<0\) for all positive \(\mu\) and \(\lambda\), we can only prove the negativity for suitable parameters.
Proposition 2. Recall the definition of \(I_{\mu,\lambda}=I_{\mu,1,\lambda}\) in 3 \[\begin{align} & I_{0,\lambda}=0, && \text{ for any \lambda\in \mathbb{R}}, \label{eqn:I950}\\ & I_{\mu,\lambda}>-\infty, && \text{ for any } \mu\geq 0 \text{ and } \lambda>0 ,\label{eqn:I95mu32is32finite}\\ & I_{\mu,\lambda}<0, && \text{ for any } (\mu,\lambda)\in \bigcup_{K\subseteq \Omega \text{ and } |K|<\infty} (0, \mu_K ] \times (\lambda_K,\infty ). \label{ineq:negativity32of32I95mu} \end{align}\] {#eq: sublabel=eq:eqn:I950,eq:eqn:I95mu32is32finite,eq:ineq:negativity32of32I95mu} where \(\mu_K=\frac{1}{\text{Vol(K)}} \int_K x_2 dx\) and \(\lambda_K=\frac{\text{Vol(K)}}{\int_K\int_K G(x,y) dydx}\).
Proof: The property ?? is trivial since \(K_0=\{0\}\). By ?? and Young’s inequality, for any \(\omega\in K_\mu\), \[\begin{align} E_{2,\lambda}[\omega]&\leq C \|x_2\omega\|_{1}^{1/2}\|\omega\|_{1}\|\omega\|_{2}^{1/2}-\frac{1}{2\lambda}\|\omega\|_2^2\\ &\leq \frac{3}{4}\Bigg(2^{-1/4}C\lambda^{1/4}\|x_2\omega\|_{1}^{1/2}\|\omega\|_{1}\Bigg)^{4/3}+\Bigg(\frac{2}{4\lambda}-\frac{1}{2\lambda}\Bigg)\|\omega\|_2^2\\ &\leq C\lambda^{1/3}\mu^{2/3}, \end{align}\] then \[I_{\mu,\lambda}=\inf_{\omega\in K_{\mu,\lambda}}\big\{-E_2[\omega ]\big\}\geq -C\lambda^{1/3}\mu^{2/3}>-\infty.\]
For any \((\mu,\lambda)\in (0,\mu_K ] \times (\lambda_K,\infty)\) where \(K\subseteq \Omega \text{ and } |K|<\infty\), define \[\begin{align} \omega_0=c_0 1_{K}, \text{ where } c_0=\mu\Big(\int_{K}x_2dx\Big)^{-1}>0. \end{align}\] It is easy to see \(\omega_0\in K_\mu\) by the choice of \(\mu\), and \[\begin{align} E_{2,\lambda}[\omega_0]&= \frac{1}{2}c_0^2\int_{K}\int_{K}G(x,y)dxdy-\frac{1}{2\lambda}c_0^2 \text{Vol}(K)\\ &=\frac{1}{2}c_0^2\Big(\int_{K}\int_{K}G(x,y)dxdy-\frac{1}{\lambda}\text{Vol}(K)\Big)\\ &>0, \end{align}\] where the last inequality is positive by the choice of \(\lambda\).
Remark 3 (For general \(\nu>0\)).
In this case, according to the scaling \(\hat{\omega}=\omega/\nu\), the ranges of \(\mu\) and \(\lambda\) in ?? should be \[\begin{align} (\mu,\lambda)\in \bigcup_{K\subseteq \Omega \text{ and } |K|<\infty} (0, \mu_K \nu] \times (\lambda_K, \infty).\label{ineq:negativity32of32I95mu4432general32nu} \end{align}\tag{11}\]
Remark 4 (Minimizing Sequence is \(L^2\) Bounded).
Any minimizing sequence \(\{\omega_n\}\) satisfying \(\omega_n\in K_{\mu_n}, \mu_n\rightarrow \mu\), and \(-E_{2,\lambda}[\omega_n]\rightarrow I_\mu\) with \(\mu, \lambda\) follows the assumption in ?? is uniformly bounded in \(L^2\). Indeed, by ?? and Young’s inequality, for arbitrary \(\epsilon >0\) and \(\omega\in K_\mu\), \[\begin{align} \frac{1}{2\lambda}\|\omega\|_{2}^2+E_{2,\lambda}[\omega]=E[\omega]\leq \frac{3}{4}\Big(\frac{C}{\epsilon^{1/2}}\|x_2\omega\|_1^{1/2}\|\omega\|_1\Big)^{4/3}+\frac{\epsilon^2}{4}\|\omega\|_2^2. \end{align}\] By taking \(\epsilon=\lambda^{-1/2}\), \[\begin{align} \|\omega\|_{2}^2\leq C\mu^{2/3}\lambda^{4/3}\|\omega\|_1^{4/3}-4\lambda E_{2,\lambda}[\omega]. \end{align}\] Thus by \(I_{\mu,\lambda}<0\), the minimizing sequence satisfies \[\begin{align} \limsup_{n\rightarrow +\infty}\|\omega_n\|_{2}\leq C\mu^{1/3}\lambda^{2/3}\limsup_{n\rightarrow +\infty}\|\omega_n\|_1^{2/3}. \end{align}\] In particular, if \(\omega\) is a minimizer, then \[\begin{align} \|\omega\|_{2}\leq C\mu^{1/3}\lambda^{2/3}\|\omega\|_1^{2/3}. \label{ineq:uniformly32bounded} \end{align}\tag{12}\]
Remark 5 (Behavior of \(\mu,\lambda\)).
In ?? , the range of \(\mu, \lambda\) is highly related to the geometric structure of \(\Omega\). For those domains with finite areas, one can directly take \(K=\Omega\) and simplify accordingly. In general, a suggestion is to let \(K\subseteq\Omega\) be a rectangle centered at \(x_0\) with proper lengths and widths of \(a\) and \(b\), respectively. Doing so, the range of \(\mu\) determined by this \(K\) can be directly calculated, and a subrange of \(\lambda\) can be obtained by using the Green’s function \(G_{a,b}\) for the rectangle and the observation \(G_{a,b}\leq G\), then a lower bound to \(\int_K\int_K G(x,y)dxdy\) can be dervied.
Proposition 6 (General structure of minimizers). Let \(\mu,\lambda\) be given by the assumption in ?? . Each minimizer \(\omega \in S_{\mu,\lambda}\) satisfies \[\label{eqn:general32structure32of32minimizer} \omega = \lambda(\psi -W x_{2} - \gamma)_+, \qquad \psi (x) = \int_{\Omega}G(x,y)\omega (y)\mathrm{d}y,\qquad{(1)}\] for some constants \(W, \gamma \in \mathbb{R}\), uniquely determined by \(\omega\).
Proof: The proof follows from a standard argument, e.g., [18], [19] for vortex rings, and is a direct consequence of Proposition 2.5 in [1], we omit the proof.
Remark 7 (Measure and \(\gamma\)). Every minimizer in \(\omega\in S_{\mu,\lambda}\) s.t. \(\gamma \neq 0\), we have \[\int \omega dx=1.\]
Proof. Exactly the same as the Remark 2.6(ii) in [1] ◻
With the preliminary estimates and the negativity of \(I_{\mu,\lambda}\) established, the next step is to prove that a general minimizing sequence actually converges to a minimizer (Theorem 9). To achieve this goal, it is necessary to study the convergence of the kinetic energy \(E[\omega]\) under weak \(L^2\) convergence.
Proposition 8 (Convergence of the kinetic energy). Let \(\mathcal{D}\) be a domain in \(\mathbb{R}^2_+\) satisfy the weak finite volume condition. Assume \(\{\omega_n\} \subset L^1 \cap L^2(\mathcal{D})\) and \(\omega \in L^2(\mathcal{D})\) satisfy:
\(\sup_n \|\omega_n\|_1 + \sup_n \|\omega_n\|_2 + \|\omega\|_1+ \|\omega\|_2\le M\) for some constants \(M\), and
\(\omega_n \rightharpoonup \omega\) weakly in \(L^2(\mathcal{D})\).
Then for any \(\epsilon>0\), there exists \(N=N(\epsilon)>0\), such that \[\begin{align} \sup_{n}\int_{D_l \cap \{|x_1|>N\}} x_2 \omega_n dx\leq \epsilon, \label{ineq:concentration32of32bdd32seq} \end{align}\qquad{(2)}\] hence \(E[\omega_n] \to E[\omega]\).
Proof: Suppose that ?? is false. Then there exists \(\epsilon_0>0\) and a subsequence \(\{\omega_{n_k}\}\) such that for any \(k \in \mathbb{Z}_{\geq 1}\), \[\begin{align} \int_{D_l \cap \{|x_1|>k\}} x_2 \omega_{n_k} dx > \epsilon_0. \end{align}\]
Thus \[\begin{align} \epsilon_0 < \int_{D_l \cap \{|x_1|>k\}} x_2 \omega_{n_k} dx \leq \Bigg(\int_{D_l \cap \{|x_1|>k\}} x_2^p dx\Bigg)^{1/p}\Bigg(\int_{D_l \cap \{|x_1|>k\}} \omega_{n_k}^q dx\Bigg)^{1/q}. \end{align}\] where \(\frac{1}{p}+\frac{1}{q}=1\) with \(p\geq 2\), we see \(q\in (1,2]\). By the weak finite volume condition 5 for \(D_l\), \[\begin{align} \int_{D_l \cap \{|x_1|>k\}} x_2^p dx \rightarrow 0, \text{ as } k \rightarrow \infty. \end{align}\] Then \[\begin{align} \|\omega_{n_k}\|_{q} \geq (\int_{D_l \cap \{|x_1|>k\}} \omega_{n_k}^q dx)^{1/q} \geq \epsilon_0 \Big(\int_{D_l \cap \{|x_1|>k\}} x_2^p dx\Big)^{-1/p}\rightarrow +\infty. \end{align}\] However, according to our assumption: \(\|\omega_{n_k}\|_{1}+ \|\omega_{n_k}\|_{2}\leq M\), then \(\|\omega_{n_k}\|_{q}\leq 2M\) for any \(q\in (1,2]\), a contradiction arises.
Based on ?? , for any \(\epsilon>0\), and define \[D_0=D_u\cup(D_l \cap \{|x_1|\leq N\}),\] where \(N=N(\epsilon^2)\) is the constant given in the previous step. By properly increasing \(N\), assume \[\begin{align} \int_{\mathcal{D}\backslash D_0} x_2\omega dx\leq \epsilon^2. \end{align}\] Then \[\begin{align} 2|E[\omega_n] & - E[\omega]| \leq 2|E[\omega_n|_{D_0}]-E[\omega|_{D_0}]| \\ & + 2\int_{\mathcal{D}}\int_{\mathcal{D}} G(x,y)\omega_n|_{D_0}(x)\omega_n|_{\mathcal{D}\backslash D_0}(y)dydx + E[\omega_n|_{\mathcal{D}\backslash D_0}]\\ & + 2\int_{\mathcal{D}}\int_{\mathcal{D}} G(x,y)\omega|_{D_0}(x)\omega|_{\mathcal{D}\backslash D_0}(y)dydx + E[\omega|_{\mathcal{D}\backslash D_0}]. \end{align}\]
For last four terms, by ?? , ?? and the fact that \(\sup_n\|x_2\omega_n|_{\mathcal{D}\backslash D_0}\|_1\leq \epsilon^2\), \(\|x_2 \omega|_{\mathcal{D}\backslash D_0}\|_1\leq \epsilon^2\), then \[\begin{align} |E[\omega_n] & - E[\omega]| \leq |E[\omega_n|_{D_0}]-E[\omega|_{D_0}]| + CM^2\epsilon. \end{align}\]
For the first term, since \(G(x,y) \in L^2(D_0 \times D_0)\) and \(\omega_n(x)\omega_n(y) \rightharpoonup \omega(x)\omega(y)\) in \(L^2(D_0 \times D_0)\), sending \(n\to\infty\) \[\begin{align} E[\omega_n|_{D_0}] = & \int_{D_0}\int_{D_0} {G}(x,y)\omega_n(x)\omega_n(y)dydx \\ & \longrightarrow \int_{D_0}\int_{D_0} {G}(x,y)\omega(x)\omega(y)dydx = E[\omega|_{D_0}]. \end{align}\]
In conclusion, as \(n\) is sufficiently large, \[\begin{align} |E[\omega_n] & - E[\omega]| \leq 2CM^2\epsilon\rightarrow 0. \end{align}\]
Proposition 8 provides a crucial ingredient: the kinetic energy is continuous with respect to weak convergence in \(L^2\) provided the sequence satisfies a uniform bound. This allows us to pass to the limit in the minimization problem, hence proving the existence of a minimizer and a general convergence theorem.
Theorem 9. Let \(\nu>0\), and \(\mu,\lambda\) satisfy 11 . For any minimizing sequence \(\{\omega_n\}\) satisfying \(\omega_n\in K_{\mu_n,\nu}\), \(\mu_n \rightarrow \mu\) and \(-E_{2,\lambda}[\omega_n] \rightarrow I_{\mu,\nu,\lambda}\), then there exists a subsequence, still denoted by \(\{\omega_n\}\), such that there exists \(\omega\in K_{\mu,\nu}\), \(\omega_n\rightarrow\omega\) and \(x_2\omega_n\rightarrow x_2\omega\) strongly in \(L^2(\mathcal{D})\) and \(L^1(\mathcal{D})\) respectively. In particular \(\omega\in S_{\mu,\nu,\lambda}\) and hence \(S_{\mu,\nu,\lambda}\neq \emptyset\).
Proof: It is sufficient to prove the case \(\nu=1\). Let \(\{\omega_n\}\) be a minimizing sequence such that \(\omega_n\in K_{\mu_n}\), \(\mu_n \rightarrow \mu\) and \(-E_{2,\lambda}[\omega_n] \rightarrow I_{\mu,\lambda}\) as \(n\rightarrow+\infty\). Since \(\{\omega_n\}\), \(\{x_2\omega_n\}\) are uniformly bounded in \(L^2\), \(L^1\) respectively, then there exists a subsequence still denoted by \(\{\omega_n\}\), \(\omega_n\rightharpoonup\omega, x_2\omega_n\rightharpoonup x_2 \omega\) in \(L^2\), \(L^1\) respectively for some \(\omega\). We shall proof that \(\omega\) is a minimizer of \(I_{\mu,\lambda}=\inf_{\omega\in K_{\mu}}\{-E_{2,\lambda}[\omega]\}\) in two steps.
Step 1: \(\omega\in K_{\mu}\).
For any \(R>0,\) since \(B(0,R)\cap \mathcal{D}\) has finite measure, then by weak convergence, \[\int_{B(0,R)\cap \mathcal{D}}\omega dx=\lim_{n\rightarrow\infty}\int_{B(0,R)\cap \mathcal{D}}\omega_n dx\leq 1,\] by the Monotone Convergence Theorem, \[\int_{\mathcal{D}}\omega dx=\lim_{R\rightarrow\infty}\int_{B(0,R)\cap \mathcal{D}}\omega dx\leq 1.\]
In addition to this, by \(x_2\omega_n\) converge weakly to \(x_2\omega\) in \(L^1\) and \(1_{\mathcal{D}}\in L^{\infty}(\mathcal{D}),\) \[\begin{align} \int x_2 \omega_n \cdot 1_{\mathcal{D}} dx=\mu_n\rightarrow \mu=\int x_2 \omega_n \cdot 1_{\mathcal{D}} dx, \end{align}\] thus \(\omega \in K_{\mu}\).
Step 2: Strong convergence and conclude the proof.
Recalling that \(\{\omega_n\}\) is uniformly bounded in \(L^2\) and \(L^1\), \(\omega_n\) converge weakly to \(\omega\) in \(L^2\), then this sequence meet the assumption in the Proposition 8, see \(E[\omega_n] \rightarrow E[\omega]\). Combining with Step 1, \[\begin{align} -I_{\mu,\lambda}=\lim_{n\rightarrow\infty}E_{2,\lambda}[\omega_n]\leq \lim_{n\rightarrow\infty}E[\omega_n]-\frac{1}{2\lambda}\liminf_{n\rightarrow\infty}\|\omega_n\|_{2}^2 \leq E_{2,\lambda}[\omega]\leq -I_{\mu,\lambda} \label{eqn:conclusion32of32claim321} \end{align}\tag{13}\]
Thus \[\begin{align} -E_{2,\lambda}[\omega] = I_{\mu,\lambda}, && \|\omega\|_{2}=\lim_{n\rightarrow\infty}\|\omega_n\|_{2}, \end{align}\] the former shows \(\omega\) is a minimizer of \(I_{\mu,\lambda}=\inf_{\omega\in K_{\mu}}\{-E_{2,\lambda}[\omega]\}\) in \(K_{\mu}\), the later show \(\omega_n\) converge to \(\omega\) strongly in \(L^2\).
In addition to that, to prove the strong \(L^1\) convergence of \(x_2\omega_n\), we separate \(\mathcal{D}\) into two subdomains: \(D_0,\) and \(\mathcal{D}\backslash D_0\), where \(D_0\) is the set used in the Proposition 8.
For the second part, by the choice of \(N\) and triangle inequality, \[\begin{align} \sup_n\int_{\mathcal{D}\backslash D_0}x_2|\omega_n-\omega|dx \leq 2\epsilon. \end{align}\]
For the first part, by \(|D_0|<\infty\) and the strong \(L^2\) convergence of \(\omega_n\), \(\omega_n|_{D_0}\) converge to \(\omega|_{D_0}\) strongly in \(L^1\). There exists a subsequence, still denoted by \(\omega_n\) s.t. \(\omega_n|_{D_0}\) converge to \(\omega|_{D_0}\) pointwisely. By Fatou’s lemma: \[\begin{align} 2\int_{D_0} x_2\omega ~ dx \leq & \liminf_n \int_{D_0} (x_2\omega_n+x_2\omega-x_2|\omega_n-\omega|)dx,\\ \leq & \mu+ \int_{D_0} x_2\omega ~ dx-\limsup_n \int_{D_0} x_2|\omega_n-\omega|dx, \end{align}\] then by the choice of \(D_0\) \[\begin{align} \mu =& \int_{D_0} x_2\omega ~ dx, +\int_{\mathcal{D}\backslash D_0} x_2\omega ~ dx\\ \leq & \mu + \epsilon-\limsup_n \int_{D_0} x_2|\omega_n-\omega|dx, \end{align}\] so that, \[\begin{align} \limsup_n \int_{D_0} x_2|\omega_n-\omega|dx\leq \epsilon. \end{align}\]
Hence, combine these two parts, \[\begin{align} \limsup_n \int_{\mathcal{D}} x_2|\omega_n-\omega|dx\leq 3\epsilon, \end{align}\] while the left hand side is independent from \(\epsilon\). Passing \(\epsilon\to 0\), we conclude \(x_2\omega_n\rightarrow x_2 \omega\) in \(L^1(\mathcal{D})\).
To obtain the same convergence of kinetic energy as [1], it is necessary to provide a strictly negative upper bound for \(I_{\mu,\lambda}\) and demonstrate that the variational problem 3 admits a minimizer. As noted in Proposition 2, the negativity of \(I_{\mu,\lambda}\) depends on finding a suitable trial function supported on a set \(K \subset \mathcal{S}\).
For our specific domain \(\mathcal{S}\), it is sufficient to consider \(K=(0, 10L) \times (0, L)\), which comfortably fits within the infinite strip \(\mathcal{S}=\mathbb{R}\times(0,L)\). The primary motivation for this choice is to obtain a concrete and strictly positive lower bound for \(-I_{\mu, \lambda}\) using the explicit Green’s function 7 and the choice of \((\mu,\lambda)\). Indeed, we take \[\begin{align} (\mu,\lambda)\in (0,L/2) \times [\frac{\pi^5}{2L^2},\infty). \label{ineq:negativity32of32I95mu4432strip} \end{align}\tag{14}\] with a trial function \[\begin{align} \omega_0=\frac{\mu}{5L^3} 1_{(0,10L)\times (0,L)}=c_0 1_{(0,10L)\times (0,L)}, \end{align}\] so that, \[\begin{align} &-I_{\mu,\lambda} \geq E_{2,\lambda }[\omega_0] \\ = & \frac{c_0^2}{2}\Bigg( \int_{(0,10L)\times (0,L)}\int_{(0,10L)\times (0,L)}G(x,y)dxdy - \frac{10L^2}{\lambda}\Bigg)\\ = & \frac{c_0^2}{2}\Bigg( \frac{1}{4\pi }\int_{K}\int_{K}\ln\Big(1+\frac{\sin(\frac{\pi}{L}x_2)\sin(\frac{\pi}{L}y_2)}{\sinh(\frac{\pi}{2L}(x_1-y_1))^2+\sin(\frac{\pi}{2L}(x_2-y_2))^2}\Big)dxdy - \frac{10L^2}{\lambda}\Bigg)\\ \geq & \frac{c_0^2}{2}\Bigg( \frac{1}{4\pi }\int_{K}\int_{K}\ln\Big(1+\sin(\frac{\pi}{L}x_2)\sin(\frac{\pi}{L}y_2)\exp(-\frac{\pi}{L}|x_1-y_1|)\Big)dxdy - \frac{10L^2}{\lambda}\Bigg). \end{align}\] By \(\ln(1+t)\geq \frac{1}{2}t\) for \(t\in(0,1)\) and \(\sin(\frac{\pi}{L}x_2)\sin(\frac{\pi}{L}y_2)\exp(-\frac{\pi}{L}(x_1-y_1))\leq 1\), \[\begin{align} -I_{\mu,\lambda}& \notag \\ \geq & \frac{c_0^2}{2}\Bigg( \frac{1}{8\pi }\Big(\int_{0}^{L}\sin(\frac{\pi}{L}x_2)dx_2\Big)^2\int_{0}^{10L}\int_{0}^{10L}\exp(-\frac{\pi}{L}|x_1-y_1|)dx_1dy_1 - \frac{10L^2}{\lambda}\Bigg) \notag \\ = & \frac{c_0^2}{2}\Bigg( \frac{1}{8\pi }\Big(\frac{2L}{\pi}\Big)^2 \Big( \frac{2L}{\pi} \cdot \frac{L}{\pi}(\exp(-10\pi)-1)+\frac{20L^2}{\pi}\Big) - \frac{10L^2}{\lambda}\Bigg), \notag \\ \geq & \frac{c_0^2}{2}\Bigg( \frac{1}{8\pi }\Big(\frac{2L}{\pi}\Big)^3 \Big( -\frac{L}{\pi}+ 10L\Big) - \frac{10L^2}{\lambda}\Bigg) \notag \\ > & 5c_0^2L^2\Bigg( \frac{3L^2}{\pi^5 } - \frac{1}{\lambda}\Bigg) \notag \\ = & \frac{5 c_0^2 L^4}{\pi^{5}}=\frac{\mu^2}{5 \pi^5 L^2}>0. \label{ineq:lower32bound32of32-I95mu4432lmabda} \end{align}\tag{15}\]
This explicit bound is essential for proving the subsequent properties, such as the propagation speed \(W\) (Corollary 4.3) and the strict decrease of \(I_{\mu, \lambda}\) (Proposition 16). With such an estimate, we can now proceed to study more properties of a potential minimizer. A fundamental step is to understand the behavior of the stream function \(\psi\), particularly its decay at infinity.
Proposition 10. For \(\omega\in L^2 \cap L^1(\mathcal{S})\) satisfying \(x_2\omega\in L^1(\mathcal{S})\) and \(\omega \geq 0 ~ (\omega \not\equiv 0)\), the stream function 2 satisfies \(\psi>0\) and \[\begin{align} \psi \rightarrow 0, \text{ as } |x_1|\rightarrow \infty, \label{eqn:decay32of32stream32function} \end{align}\qquad{(3)}\] as well as \[\begin{align} E[\omega]=\frac{1}{2}\|\nabla\psi\|_2^2. \label{eqn:kinetic32energy32and32stream32fct} \end{align}\qquad{(4)}\]
Proof. ?? can be conclude by the observation \(0<\psi\leq \psi^H\) in \(S\) and \(\psi^H\rightarrow 0\) as \(|x|\rightarrow \infty\) (see the Proposition 2.2 in [1]).
As for the ?? , we take a non-increasing function \(\theta \in C_C[0,\infty)\) satisfying \(\theta=1\) in \([0,1]\), vanishes in \([2,\infty)\) and set the cut-off function by \(\theta_R(x)=\theta(|x|/R)\) in \(\mathcal{S}\). Since \(-\Delta \psi = \omega\) in \(\mathcal{S}\) and \(\psi(x_1,0)=\psi(x_1,L)=0\), by multiplying \(\psi \theta_R\) by \(-\Delta \psi = \omega\) and integration by parts, \[\begin{align} \int_{\mathcal{S}}\psi \omega \theta_R=\int_\mathcal{S}|\nabla\psi|^2\theta_R -\frac{1}{2}\psi^2\Delta \theta_R. \end{align}\] Since \(\psi \rightarrow 0\) as \(|x_1|\rightarrow 0\) by ?? , the second term vanishes as \(R\rightarrow \infty\). Hence ?? follows from the monotone convergence theorem. ◻
As in [1], the positivity of \(W\) will be shown to imply compactness of support for minimizers. To demonstrate it, it is necessary to study the regularity of \(\psi\). We denote by \(\operatorname{BUC}(\mathcal{S})\) the space of all bounded uniformly continuous functions in \(\mathcal{S}\). For an integer \(k\geq 0, \operatorname{BUC}^{k+\alpha}(\mathcal{S})\) denotes the space of all \(\psi \in \operatorname{BUC}(\mathcal{S})\) such that \(\partial_x^l \psi \in \operatorname{BUC}(\mathcal{S}) \cap C^\alpha(\mathcal{S})\), for \(|l| \leq k\).
Proposition 11. For any \((\mu,\lambda)\) given in 14 , \(\omega \in S_{\mu,\lambda}\), the stream function satisfies \(\psi \in \operatorname{BUC}^{2+\alpha}(\overline{\mathcal{S}})\), \(0<\alpha<1\), \(\psi/x_2 \in \operatorname{BUC}^{1+\alpha}(\mathcal{S})\), and \[\begin{align} \frac{\psi(x)}{x_2} \to 0 \quad \text{as } |x_1| \to \infty.\label{eqn:psi47x232decay} \end{align}\qquad{(5)}\]
Proof. The proof is standard and follows the strategy of Proposition 2.7 [1]. Since \(\omega \in L^1 \cap L^2(\mathcal{S})\), the Biot-Savart law and standard elliptic estimates imply \(\nabla^2 \psi \in L^q(\mathcal{S})\) for \(q \in (1,2)\) and \(\nabla \psi \in L^p(\mathcal{S})\) with \(1/p = 1/q - 1/2\). By the structure equation ?? and the Lipschitz continuity of \(f(t)=t_+\), we obtain \(\partial_x^l \psi \in L^p_{\mathrm{ul}}(\overline{\mathcal{S}})\) for \(|l|=3\). The Sobolev embedding then yields \(\psi \in \mathrm{BUC}^{2+\alpha}(\overline{\mathcal{S}})\). The claim \(\psi/x_2 \in \mathrm{BUC}^{1+\alpha}(\overline{\mathcal{S}})\) follows from \(\psi(x_1,0)=0\) and the identity \[\frac{\psi(x_1,x_2)}{x_2} = \int_0^1 (\partial_2\psi)(x_1, x_2 s)\,ds.\] Finally, ?? follows from [prop:kinetic32energy32and32stream32function4432and32decay] and Hardy’s inequality as Proposition 2.7 [1]. ◻
The decay property ?? is crucial for analyzing the structure equation ?? . Since a minimizer \(\omega\) must vanish at infinity, the observation implies the sign of \(\gamma\) and then \(W\).
Corollary 12 (\(\gamma=0\) and \(W>0\) for small \(\mu\)). Let \(\mu,\lambda\) given by 14 , there exists a constant \(M_1 > 0\) such that if \(0 < \mu \le M_1\), then every minimizer \(\omega \in S_{\mu,\lambda}\) satisfies \[\begin{align} \int_{\mathcal{S}}\omega \,dx < 1 \label{ineq:L132norm32of32minimizer32is32strictly32smaller32that321}. \end{align}\qquad{(6)}\] In particular, \(\gamma = 0\) by Remark 7 and then \(W>0\).
Proof. Recall the structure of \(\omega\), ?? . Since \(\int_{\mathcal{S}}\omega<\infty\), then \((\psi-Wx_2-\gamma)_+=\frac{1}{\lambda}\omega\rightarrow 0\) as \(|x_1|\rightarrow\infty\). Consider a sequence \(x_n=(x_{n,1},x_{n,2})\) such that \(x_{n,1}\rightarrow \infty\) and \(x_{n,2} \rightarrow 0\), then combine with the decay of \(\psi\), ?? , \[\begin{align} \limsup_n ( \psi(x_n)-Wx_{2,n}-\gamma )\leq 0, \end{align}\] this implies \(\gamma\geq 0\) for all \(\mu\). As for \(W\), by taking an another sequence \(x_n=(x_{n,1},x_{n,2})\) such that \(x_{n,1}\rightarrow \infty\) and \(x_{n,2} \rightarrow L\), then \(-WL-\gamma \leq 0\), so that \(W\geq -\gamma/L\).
To show \(\gamma=0\), we noticed that: if \(W<0\), then \(-W x_2-\gamma \leq -WL-\gamma\leq 0\) implies \(\omega=\lambda(\psi-Wx_2-\gamma)_+\leq \lambda \psi\) for any \(x_2\in (0,L)\). Such inequality also holds when \(W\geq 0\). Thus, we can use 10 to obtain \[\begin{align} \int_{0<x_2<2\mu}\omega dx\leq & \lambda \int_{0<x_2<2\mu} \psi dx\leq \lambda \int_{0<x_2<2\mu} \psi^H dx,\\ \leq & \lambda \int_{0<x_2<2\mu} \int_{\mathbb{R}^2_+} G_H(x,y)\omega(y) dy dx, \end{align}\] and all remaining steps are followed by Remark 2.6 (iii) in [1]
We shall finish the proof by deriving a contradiction. Suppose not, i.e. \(W=0\), then \(\psi \in L^2(S)\) is a solution to the system: \[\begin{align} \left\{ \begin{aligned} -\Delta\psi=\omega=\lambda(\psi)_+&=\lambda \psi && \text{ in } \mathcal{S},\\ \psi&=0 && \text{ on } \partial\mathcal{S},\\ \psi &\rightarrow 0 && \text{ as } |x_1|\rightarrow \infty. \end{aligned} \right. \end{align}\] Note that \(\psi \in \operatorname{BUC}^{2+\alpha}(\overline{\mathcal{S}})\), then it is possible to do a Fourier transform of \(\psi\) in \(x_1\)-variable, we define \[\widehat{\psi}(\xi,x_2) := \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} e^{-i\xi x_1}\,\psi(x_1,x_2)\,dx, \qquad \xi\in\mathbb{R},\; x_2\in(0,L).\] By Plancherel’s theorem, \(\widehat\psi\in L^2(\mathcal{S})\) and \(\|\psi\|_{L^2(\mathcal{S})} = \|\widehat\psi\|_{L^2(\mathcal{S})}\).
Applying the Fourier transform (in \(x_1\)) to the equation \(-\Delta\psi = \lambda\psi\) gives \[\begin{align} \label{eq:transverse} \left\{\begin{aligned} -\partial_{x_2}^2\widehat\psi &= (\lambda-\xi^2)\,\widehat\psi,\\ \widehat\psi(\xi,0) &= \widehat\psi(\xi,L) = 0\qquad \text{for any }\xi\in\mathbb{R}. \end{aligned}\right. \end{align}\tag{16}\] For each fixed \(\xi\), 16 is a problem for the unknown function \(y\mapsto\widehat\psi(\xi,y)\). By solving the elementary single variable boundary valued ODE, \(\widehat\psi(\xi,\cdot)\) has a non‑zero solution exists if and only if \[\lambda - \xi^2 = \nu_m,\qquad \nu_m := \left(\frac{m\pi}{L}\right)^2,\; m=1,2,3,\dots\] More precisely:
If \(\lambda-\xi^2 \neq \nu_m\) for all \(m\), then \(\widehat\psi(\xi,\cdot)\equiv 0\) on \([0,L]\).
If \(\lambda-\xi^2 = \nu_m\) for some \(m\), then \(\widehat\psi(\xi,y) = c(\xi)\,\sin\!\bigl(\frac{m\pi y}{L}\bigr)\)for some coefficient \(c(\xi)\).
Consequently, the set of \(\xi\) for which \(\widehat\psi(\xi,\cdot)\) can be non‑zero is contained in \[E := \bigcup_{m\in\mathbb{N}} \Bigl\{\xi\in\mathbb{R} : \xi^2 = \lambda - \Bigl(\frac{m\pi}{L}\Bigr)^2\Bigr\}.\] For a fixed \(\lambda\), only finitely many \(m\) satisfy \(\lambda \ge (m\pi/L)^2\); for each such \(m\) there are at most two values of \(\xi\), namely \(\pm\sqrt{\lambda-(m\pi/L)^2}\). Thus, \(E\) is a finite set with Lebesgue measure zero, so that \(\|\psi\|_{L^2(\mathcal{S})} = \|\widehat\psi\|_{L^2(\mathcal{S})}=0\) and then \(\psi\equiv 0\).
By the structure analysis ?? , \(\omega=0\), hence \(0>-I_{\mu,\lambda}=E_2[\omega]=0\). This yields a contradiction. ◻
Then combine ?? with \(\gamma=0\) and \(W>0\), we have
Proposition 13. Under the assumption in the Proposition 11, \(\operatorname{spt}\omega\) is compact in \(\overline{\mathbb{R}_+^2}\).
Proof. Since \(\operatorname{spt}\omega = \overline{\{x \in S \mid \psi(x) - W x_2 - \gamma > 0\}}\) for \(W > 0\) and \(\gamma \ge 0\), \[\begin{align} W x_2 \le \psi(x), \qquad x \in \operatorname{spt}\omega. \end{align}\] Because \(\psi/x_2 \to 0\) as \(|x_1| \to \infty\) by ?? , the assertion follows. ◻
Proposition 14 (Steiner symmetrization). For \(\omega \ge 0\) satisfying \(\omega \in L^2 \cap L^1(\mathcal{S})\) and \(x_2\omega \in L^1(\mathcal{S})\), there exists \(\omega^* \ge 0\) such that \[\begin{align} \begin{aligned} &\omega^*(x_1,x_2) = \omega^*(-x_1,x_2), \\ &\omega^*(x_1,x_2) \text{ is non-increasing for } x_1 > 0. \end{aligned} \end{align}\] Moreover, \[\begin{align} \begin{aligned} \|\omega^*\|_q &= \|\omega\|_q, \quad 1\le q \le 2, \\ \|x_2\omega^*\|_1 &= \|x_2\omega\|_1, \\ E[\omega^*] &\ge E[\omega]. \end{aligned} \end{align}\]
Proof. Recall the structure of the Green’s function for \(\mathcal{S}\), 7 . For any fixed \(x_2,y_2\), \(G\) is a strictly decreasing function with respect to \(|x_1-y_1|\). Then the Riesz Rearrangement inequality can be applied, which directly implies this theorem. See [20], Appendix I, and [21], p.1053. ◻
Proposition 15. Let \(A,R \ge 1, Q = \{x \in \mathcal{S}\mid |x_1| < AR,\; x_2 < R\}\). Let \(\psi\) be the stream function 2 for \(\omega \in L^2\cap L^1(\mathcal{S})\) satisfying \(x_2\omega \in L^1(\mathcal{S})\) and \(\omega \ge 0\). Assume that 9 holds for \(\omega\). Then, \[\begin{align} & \psi(x)\leq C x_2^{1/2}\|\omega\|_1^{1/2}\|\omega\|_2^{1/2} + \|\omega\|_1 +x_2\Big(\frac{A}{x_1}\Big)^2\|x_2 \omega\|_1 , x_2 \leq \frac{|x_1|}{A},\label{enq:upperbound32estimate}\\ & \int_{\mathcal{S}\setminus Q} \psi(x)\omega(x)\,dx \leq \frac{C}{\min\{A,R\}^{1/2}} \bigl( \|\omega\|_{L^1\cap L^2}^2 + \|x_2\omega\|_{L^1}^2 \bigr).\label{enq:far-field32estimate} \end{align}\] {#eq: sublabel=eq:enq:upperbound32estimate,eq:enq:far-field32estimate} The constant \(C\) is independent of \(\omega\) and \(A, R \ge 1\).
Proof. Combine the observation \(0\leq \psi\leq \psi^H\) on \(\mathcal{S}\) and the Proposition 3.3, 3.4 in [1], then the result conclude. ◻
Proposition 16 (Convergence of the kinetic energy). Assume that the sequence \(\{\omega_n\} \subset L^1 \cap L^2(\mathcal{S})\) satisfies the following:
\(\sup_n \|\omega_n\|_1 + \sup_n \|\omega_n\|_2 \leq M\) for some constants \(M\), and
\(\omega_n \rightharpoonup \omega\) in \(L^2(\mathcal{S})\).
If each \(\omega_n\) is nonnegative and satisfies 9 , then \(E[\omega_n] \to E[\omega]\) as \(n \rightarrow \infty\).
Proof. We decompose the energy into two terms \[\begin{align} 2E[\omega_n] = \int_{\mathcal{S}} \psi_n(x)\omega_n(x)\,dx = \int_{Q} \psi_n(x)\omega_n(x)\,dx + \int_{\mathcal{S}\setminus Q} \psi_n(x)\omega_n(x)\,dx, \end{align}\] and observe that \[\begin{align} \int_{Q} \psi_n(x)&\omega_n(x)dx \\ &= \int_{Q} \omega_n(x) \int_{Q} G(x,y)\omega_n(y)\,dydx + \int_{Q} \omega_n(x)\int_{\mathcal{S}\setminus Q} G(x,y)\omega_n(y)dydx,\\ &= \int_{Q} \omega_n(x) \int_{Q} G(x,y)\omega_n(y)\,dydx + \int_{\mathcal{S}\setminus Q} \omega_n(y)\int_{Q} G(x,y)\omega_n(x)dxdy,\\ &\leq \int_{Q} \omega_n(x) \int_{Q} G(x,y)\omega_n(y)\,dydx + \int_{\mathcal{S}\setminus Q} \psi_n(x)\omega_n(x)\,dx. \end{align}\] Applying ?? yields \[\begin{align} \Bigl|E[\omega_n] - \frac{1}{2}\int_{Q}\int_{Q} G(x,y)\omega_n(x)\omega_n(y)\,dx\,dy \Bigr| \leq & \int_{\mathcal{S}\setminus Q} \psi_n(x)\omega_n(x)\,dx, \\ \leq & \frac{C}{\min\{A,R\}^{1/2}}. \end{align}\] By estimating \(E[\omega]\) in the same way, \[\begin{align} 2|E[\omega_n] - E[\omega]| \le \Bigl| \int_{Q}\int_{Q} G(x,y)(\omega(x)\omega(y) - \omega_n(x)\omega_n(y))\,dx\,dy \Bigr| + \frac{C}{\min\{A,R\}^{1/2}}. \end{align}\] Since \(G(x,y) \in L^2(Q \times Q)\) and \(\omega_n(x)\omega_n(y) \rightharpoonup \omega(x)\omega(y)\) in \(L^2(Q \times Q)\), sending \(n\to\infty\) and \(A,R\to\infty\) imply the desired result. ◻
Before stating the next result, we pause to note that the path to Lemma 3.2 in [1] has now been fully cleared in our strip setting. Specifically, Proposition 14 provides the Steiner symmetrization with proper properties, Proposition 16 guarantees that the weak \(L^2\) convergence of the kinetic energy for symmetrized sequences, and Corollary 12 ensures the strict positivity of the propagation speed \(W\). With these three components in place, the variational construction in Lemma 3.2 [1] carries over verbatim.
Proposition 17. For \(0 < \nu < \infty, 0< \mu <M_1 \nu, \lambda > \frac{\pi^5}{2L^2}\), there exists a minimizer \(\tilde{\omega}\in \tilde{K}_{\mu,\nu}\) such that \[\begin{align} E_{2,\lambda}[\tilde{\omega}]=\inf_{\tilde{\omega} \in \tilde{K}_{\mu,\nu}} \left\{-E_{2,\lambda}[\tilde{\omega}]\right\}, \end{align}\] where \(\tilde{K}_{\mu,\nu} = \Bigl\{ \omega \in L^2 \cap L^1(S) \;\Big|\; \omega \ge 0,\; \int_{S} x_2\omega \,dx \le \mu,\; \int_{S} \omega \,dx \le \nu \Bigr\}.\) Moreover, \(\int x_2\omega =\mu\) with a compact support. In particular, it is also a minimizer of 3 in \({K}_{\mu,\nu}\).
Proof. The result follows by reproducing the proof of Lemma 3.2 in [1] in our setting. That argument rests on two pillars: i) a Steiner symmetrization that preserves the constraints while raising the penalized energy (given by Proposition 14), and ii) the continuity of the kinetic energy with respect to weak-\(L^2\) convergence for a symmetrized sequence (given by Proposition 16).
In the final step, one can use the same argument in Lemma 3.2 in [1] to prove \(\int x_2\omega =\mu\). This implies that such a maximizer in the enlarged space \(\tilde{K}_{\mu,\nu}\) is actually a maximizer in the admissible space \(K_{\mu,\nu}\). ◻
Having established the existence of a minimizer, we now aim for the main result of this section: proving that any minimizing sequence is (up to a subsequence) strongly convergent to a minimizer. The concentration-compactness principle (Proposition 22) will be the primary tool. However, to apply this principle and, in particular, to exclude the dichotomy scenario, the strict decrease of \(I_{\mu, \lambda}\) with respect to \(\mu\) is required. We begin by deriving important uniform bounds for all minimizers with small impulse. These bounds, especially the \(L^1\) bound, are indispensable for the perturbation argument that proves the strict decreasing (Proposition 21).
Proposition 18. (\(L^1\) estimate of minimizers)For any \(0<\nu<\infty, 0<\mu < M_1 \nu\) and \(\lambda \geq \pi^5(2 L^2)^{-1}\), we have \[\begin{align} \sup_{\omega\in S_{\mu,\nu,\lambda }} |\operatorname{spt}\omega|\leq \frac{5}{2}\pi^5 \nu^{-2} L^2. \label{ineq:minimizer32support32estimate} \end{align}\qquad{(7)}\] Consequently, \[\begin{align} \sup_{\omega\in S_{\mu,\nu,\lambda }}\| \omega \|_1\leq C^{3} \mu\nu^{-3}\lambda^{2} L^3, \label{ineq:minimizer32L132estimate}\\ \sup_{\omega\in S_{\mu,\nu,\lambda }}\| \omega \|_2\leq C^{3} \mu\nu^{-2}\lambda^{2} L^2, \label{ineq:minimizer32L232estimate} \end{align}\] {#eq: sublabel=eq:ineq:minimizer32L132estimate,eq:ineq:minimizer32L232estimate} for some constant \(C\) independent from \(\mu,\nu,\lambda,L\).
Proof. It is sufficient to consider the case when \(\nu=1\). For any \(\omega\in S_{\mu,\lambda}\), with \(\mu\leq M_1\) and \(\lambda \geq \pi^5(2 L^2)^{-1}\). Recall the equation ?? with \(W=W(\omega)>0\) and \(\gamma=0\). Then, multiplying the equation \(-\Delta(\psi-W x_2)=\lambda(\psi-W x_2)_+\) by \(\psi-Wx_2\) and integrating by parts, \[\begin{align} \int_{\operatorname{spt}\omega} |\nabla \psi |^2-2W \partial_2\psi +W^2dx = & \int_{\operatorname{spt}\omega} |\nabla (\psi-Wx_2)|^2dx,\\ = &\lambda \int_{\operatorname{spt}\omega}|\psi -Wx_2|^2dx=\frac{1}{\lambda} \int_{\operatorname{spt}\omega}|\omega|^2dx. \end{align}\]
Since \(\psi-W x_2=0\) on \(\partial( \operatorname{spt}\omega )\), we have \[\begin{align} \int_{\operatorname{spt}\omega} \partial_2\psi = & \int_{\operatorname{spt}\omega} \partial_2(\psi-Wx_2) + W|\operatorname{spt}\omega|, \\ = & \int_{\partial( \operatorname{spt}\omega )} n_2(\psi-Wx_2) + W|\operatorname{spt}\omega|=W|\operatorname{spt}\omega|. \end{align}\] Then \[\begin{align} 2E[\omega] - W^2|\operatorname{spt}\omega| & \geq \int_{\operatorname{spt}\omega} |\nabla \psi |^2-W^2|\operatorname{spt}\omega|= \frac{1}{\lambda} \int_{\operatorname{spt}\omega}|\omega|^2dx. \end{align}\] Hence, \[\begin{align} W^2|\operatorname{spt}\omega| & \leq 2E_{2,\lambda }[\omega]\leq -2I_{\mu,\lambda}. \end{align}\] Meanwhile, \[\begin{align} \begin{aligned} -I_{\mu,\lambda} = & E_{2,\lambda}[\omega]=\frac{1}{2}\int_{\operatorname{spt}\omega} \psi \omega -\frac{1}{2\lambda}\int_{\operatorname{spt}\omega} \omega^2, \\ = & \frac{1}{2}\int_{\operatorname{spt}\omega} (\psi-Wx_2) \omega +\frac{1}{2}W\mu -\frac{1}{2\lambda}\int_{\operatorname{spt}\omega} \omega^2=\frac{1}{2}W\mu, \end{aligned} \label{eqn:value32of32I95mu32and32W} \end{align}\tag{17}\] then \[\begin{align} |\operatorname{spt}\omega| & \leq \mu/W. \end{align}\] Recall there exists a lower bound for \(-I_{\mu,\lambda} \geq \frac{\mu^2}{5 \pi^5 L^2}\), then the identity 17 implies \[\begin{align} W\geq \frac{2 \mu}{5 \pi^5 L^2}. \end{align}\] Hence \[\begin{align} |\operatorname{spt}\omega| & \leq \mu/W \leq \frac{5}{2}\pi^5 L^2, \end{align}\] and the proof finished by the Cauchy inequality and the \(L^2\) uniform bound 12 while \(\omega_n\equiv \omega\) for all \(n\): \[\begin{align} \|\omega\|_1\leq & \|\omega\|_{2}|\operatorname{spt}\omega|^{1/2}\leq C\mu^{1/3}\lambda^{2/3}L\|\omega\|_1^{2/3},\\ \Rightarrow \|\omega\|_1 \leq & C^3\mu \lambda^{2}L^3,\\ \Rightarrow \|\omega\|_2 \leq & C\mu^{1/3}\lambda^{2/3}\|\omega\|_1^{2/3}\leq C^3\mu\lambda^{2}L^2. \end{align}\] ◻
Remark 19. The proof shows that, for a small \(\mu\), the parameter \(W\) is independent of the choice of \(\omega\), i.e., \(W=-2I_{\mu,\lambda}/\mu\).
Remark 20. Combined with estimates ?? , ?? , and ?? , together with the lower bound 15 , one can confirm \(-I_{\mu,\lambda}\in\mathcal{O}(\mu^2)\) as \(\mu\) is sufficiently small.
Before proving the strict monotonicity, we recall that in Proposition 2 and the analysis for 14 we constructed an explicit trial function \(\omega_0 = c_0 \mathbf{1}_{(0,10L)\times(0,L)}\) supported on a fixed compact set \(K\subset\mathcal{S}\) with \(E_{2,\lambda}[\omega_0] > 0\). This function serves as a “seed” for perturbing minimizers without violating the mass constraint, thanks to the uniform \(L^1\) bound provided by Proposition 18.
Proposition 21. (Strictly decreasing of \(I_\mu\))For any \(0<\nu<\infty\) and \(\lambda \geq \pi^5(2 L^2)^{-1},\) take \(M_2=\min\{C^{-3}\nu^3 \lambda^{-2}L^{-3},M_1\nu\}>0\), then for any \(0<\alpha<\mu<M_2\), \(I_{\alpha,\nu,\lambda} > I_{\mu,\nu,\lambda}\).
Proof. It is sufficient to consider the case \(\nu=1\). For any \(0<\mu <M_2\), fix a \(\omega_\mu\in S_{\mu,\lambda}\). Note that \(\omega_\mu\) is compactly supported has an \(L^1\) upper bound \(C^3 \mu\lambda^{2} L^3\) and recall the function \(\omega_0\) given in Proposition 2. For \(s\in(0,1-C^3 \mu\lambda^{2} L^3)\), we define \(\omega_{s}\in K_{\mu(1+s)}\) by \[\begin{align} \omega_s(x)=\omega_{\mu}(x)+s\omega_0(x+\tau e_1), \end{align}\] where \(\tau\) is sufficiently large constant such that \(\operatorname{spt}(\omega_\mu) \cap \operatorname{spt}(\omega_0(\cdot+\tau e_1))=\emptyset\). Then for any \(\alpha\in (\mu,~\mu(1+1-C^3 \mu\lambda^{2} L^3) ),\) \[\begin{align} -I_{\alpha, \lambda} & \geq E_{2,\lambda}[\omega_s]=E_{2,\lambda}[\omega_\mu]+s^2 E_{2,\lambda}[\omega_0]+2s\int_S\int_S G(x,y)\omega_\mu(x)\omega(y+se_1)dxdy,\\ & > -I_{\mu,\lambda}+\frac{1}{5 \pi^5 L^2}\mu^2s^2,\\ & =-I_{\mu,\lambda}+\frac{1}{5 \pi^5 L^2}(\alpha-\mu)^2. \end{align}\] Here, \(s=\alpha/\mu -1\) and \(\alpha=\mu(1+s)\). Then \(\mu\) induce a monotone increasing and bounded above sequence \(\{ \mu_k \}_{k}\) by \(\mu_0=\mu, \mu_{k+1}=\mu_k(1+1-C^3 \mu_k\lambda^{2} L^3)\) with a limit \(C^{-3}\lambda^{-2}L^{-3}\). Hence, for any \(\mu<\alpha<M_2\), there exists some \(k\geq 0\) such that \(\alpha\in(\mu_k,\mu_{k+1}]\), then \(I_{\alpha,\lambda}<I_{\mu_k,\lambda}\leq I_{\mu,\lambda}\). ◻
Proposition 22. Let \(0<\mu<\infty\). Let \(\{\rho_n\}\subset L^1(\mathcal{S})\) satisfy \[\begin{align} \rho_n\ge 0,\quad n\ge 1,\qquad \int_{S}\rho_n\,dx = \mu_n \to \mu \quad\text{as } n\to\infty . \end{align}\] There exists a subsequence \(\{\rho_{n_k}\}\) satisfying one of the following:
(Compactness) There exists a sequence \(\{y_k\}\subseteq\overline{\mathcal{S}}\) such that \(\rho_{n_k}(\cdot+y_k)\) is tight, i.e., for every \(\epsilon>0\) there exists \(R>0\) such that \[\begin{align} \liminf_{k\to\infty}\int_{B(y_k,R)\cap \mathcal{S}}\rho_{n_k}\,dx \ge \mu-\epsilon. \end{align}\] In addition, it is possible to assume \(y_{k,2}=0\) for all \(k\).
(Vanishing) For each \(R>0\) \[\begin{align} \lim_{k\to\infty}\sup_{y\in \mathcal{S}}\int_{B(y,R)\cap \mathcal{S}}\rho_{n_k}\,dx = 0 . \end{align}\]
(Dichotomy) There exists \(\alpha\in(0,\mu)\) such that for every \(\epsilon>0\) there exist \(k_0\ge 1\) and \(\{\rho_k^1\},\{\rho_k^2\}\subset L^1(\mathcal{S})\) such that \(\operatorname{spt}\rho_k^1\cap\operatorname{spt}\rho_k^2=\emptyset\), \(0\le\rho_k^i\le\rho_{n_k}\) for \(i=1,2\), and \[\begin{align} \limsup_{k\to\infty}\Bigl\{\|\rho_{n_k}-\rho_k^1-\rho_k^2\|_{L^1} + \Bigl|\int_{S}\rho_k^1\,dx-\alpha\Bigr| + \Bigl|\int_{S}\rho_k^2\,dx-(\mu-\alpha)\Bigr|\Bigr\} \le \epsilon , \end{align}\] \[\begin{align} d(\operatorname{spt}\rho_k^1,\operatorname{spt}\rho_k^2) \to \infty \quad\text{as } k\to\infty . \end{align}\]
Proof. The assertion is proved by Lemma I.1 in [22] for \(\mathbb{R}^2\) and the fixed mass \(\mu_n=\mu\) by using Lévy’s concentration function. The case of \(\mu_n \rightarrow \mu\) is proved by the Lemma 4.1 in [1] for \(\mathbb{R}_2^+\). This proof is a direct consequence of result in [1] by view \(L^i(\mathcal{S})=\{\omega\in L^i(\mathbb{R}_+^2) ~ | ~ \operatorname{spt}\omega\subseteq \overline{\mathcal{S}} \}\). From this perspective, the sequence of \(\rho_n\) meets the requirement of Lemma 4.1 in [1] and then yields three possibilities: compactness, vanishing, or dichotomy.
(Compactness) In this case, the tightness is originally stated with centres \(y_k \in \mathbb{R}^2_+\). If \(\limsup_k y_{k,2}=\infty\), then for large \(k\), the ball \(B(y_k, R)\) would not intersect \(\mathcal{S}\), forcing the integral to vanish and contradicting tightness. Therefore, \(\sup_k y_{k,2} < \infty\). By replacing original \(R\) by \(R+\limsup_k y_{k,2}\), it is possible to assume \(y_{2,k}=0\) for all \(k\), so that \(y_k \in \overline{\mathcal{S}}\) for all \(k\).
(Vanishing) In this case, one can use the set relation \(\mathcal{S}\in \mathbb{R}^2_+\) to conclude for any \(R>0\), \[\begin{align} 0\leq \lim_{k\to\infty}\sup_{y\in \mathcal{S}}\int_{B(y,R)\cap \mathcal{S}}\rho_{n_k}\,dx \leq \lim_{k\to\infty}\sup_{y\in \mathbb{R}_+^2}\int_{B(y,R)\cap \mathbb{R}_+^2}\rho_{n_k}\,dx = 0 . \end{align}\]
(Dichotomy) In this case, due to \(0\le\rho_k^i\le\rho_{n_k}\), then \(\operatorname{spt}\rho_{n_k}^i\subseteq \operatorname{spt}\rho_k \subseteq \overline{\mathcal{S}}\) for \(i=1,2\), so that \(\{\rho_k^1\},\{\rho_k^2\}\subset L^1(\mathcal{S})\) and conclude the result.
◻
Under the strict decreasing of \(I_{\mu,\lambda}\) (Proposition 21), the convergence of kinetic energy for Steiner-symmetrized sequences (Proposition 16), and the concentration-compactness alternatives (Proposition 22), we can now outline the proof of the general convergence theorem for the strip. This theorem parallels the strategy of Abe & Choi [1] for the half-plane, but its validity in the strip domain \(\mathcal{S}\) hinges on the adaptations made in the preceding propositions.
The proof proceeds by eliminating vanishing and dichotomy for a minimizing sequence \(\{\omega_n\}\). Vanishing is ruled out by the negativity of \(I_{\mu,\lambda}\). The more delicate case is dichotomy, where a minimizing sequence splits into two well-separated components. By applying Steiner symmetrization to these components and using the strict monotonicity of \(I_{\mu,\lambda}\), we deduce a contradiction to its minimality, forcing compactness instead. We now formalize this result.
Theorem 23. Let \(0<\nu<\infty, \mu<M_2 \nu\) and \(\lambda \geq \pi^5(2 L^2)^{-1}\). For any minimizing sequence \(\{\omega_n\}\) satisfying \(\omega_n\in K_{\mu_n,\nu}\), \(\mu_n \rightarrow \mu\) and \(-E_{2,\lambda}[\omega_n] \rightarrow I_{\mu,\nu,\lambda}\), then there exists a subsequence, still denoted by \(\{\omega_n\}\), s.t. there exists \(\omega\in K_{\mu,\nu}\), \(\omega_n\rightarrow\omega\) and \(x_2\omega_n\rightarrow x_2\omega\) strongly in \(L^2(\mathcal{S})\) and \(L^1(\mathcal{S})\) respectively.
Proof. The proof follows the concentration-compactness argument of Theorem 1.3 in [1], which we briefly outline. By Proposition 22, the sequence \(x_{2}\omega_{n}\) (up to a subsequence) satisfies one of three scenarios: compactness, vanishing, or dichotomy. Vanishing cannot occur since \(I_{\mu,\lambda} < 0\). If dichotomy were to occur, the sequence would split into two non-trivial parts with strictly smaller impulses. Applying Steiner symmetrization (Proposition 16) and the strict monotonicity of \(I_{\mu,\lambda}\) (Proposition 21), one derives a contradiction to the minimizing property of \(\{\omega_{n}\}\), as shown in detail in the Theorem 1.3, [1]. Therefore, the sequence must be locally compact up to horizontal translations along the boundary \(\mathcal{S}\). The strong convergence of the kinetic energy then follows from Proposition 16, which identifies the limit as a minimizer. ◻
Having established the existence of minimizers and a general convergence theorem, we now turn to their orbital stability with \(C_c^{\infty}(\Omega)\) initial data. Using the strong convergence shown in Theorem 9 for \(\Omega=\mathcal{D}\) and Theorem 23 for \(\Omega=\mathcal{S}\), we can show that the set of minimizers is orbital stable in the sense that any flow starting close to a minimizer will remain close to the set of minimizers during its lifespan. We conclude this section by deriving the precise functional form of these minimizers.
The proof here is the same as that for Theorem 1.4 in [1], but to ensure completeness, we present it below.
Theorem 24. Let \(\nu,\mu,\lambda\) given in the Theorem 9 for the case \(\Omega=\mathcal{D}\) and given in the Theorem 23 for the case \(\Omega=\mathcal{S}\), then \(S_{\mu,\nu,\lambda}\) is orbitally stable in the sense that: for any \(\epsilon>0\), there exists \(\delta>0\) s.t. \(\forall ~ \xi_0\in C_c^{\infty}(\Omega), ~ x_2\xi_0\in L^1(\Omega), ~ \xi_0\geq0, ~ \|\xi_0\|\leq \nu\) and \[\begin{align} \inf_{\omega\in S_{\mu,\nu,\lambda}}\Big\{ \|\xi_0-\omega\|_{2}+\|x_2(\xi_0-\omega)\|_{1} \Big\}\leq \delta, \end{align}\] the solution \(\xi(t)\) satisfy: \(\forall ~ t \in [0,T_{\text{lifespan}})\), \[\begin{align} \inf_{\omega\in S_{\mu,\nu,\lambda}}\Big\{ \|\xi(t)-\omega\|_{2}+\|x_2(\xi(t)-\omega)\|_{1} \Big\}\leq \epsilon. \label{eqn:stable32result} \end{align}\qquad{(8)}\]
Proof: It suffices to prove the case when \(\nu=1\). Suppose that ?? is false. Then there exists \(\epsilon_0>0\), a sequence \(\{\xi_{0,n}\}\subseteq C_c^\infty(\mathcal{D}), ~ \xi_{0,n} \geq 0, ~ \|\xi_{0,n}\|_1\leq 1\) and \(t_n\) in the lifespan of the solution \(\xi_n\) based on the initial data \(\xi_{0,n}\) s.t. \[\begin{align} \inf_{\omega\in S_{\mu,\lambda}}\Big\{ \|\xi_{0,n}-\omega\|_{2}+\|x_2(\xi_{0,n}-\omega)\|_{1} \Big\} & \leq 1/n, \\ \inf_{\omega\in S_{\mu,\lambda}}\Big\{ \|\xi_n(t_n)-\omega\|_{2}+\|x_2(\xi_n(t_n)-\omega)\|_{1} \Big\} &> \epsilon_0. \end{align}\] We write \(\xi_n=\xi_n(t_n)\) by suppressing \(t_n\). We take \(\omega_n\in S_{\mu,\lambda}\) such that \(\|\xi_{0,n}-\omega\|_{2}+\|x_2(\xi_{0,n}-\omega)\|_{1}\rightarrow 0\). By ?? , \[| E_{2,\lambda}[\xi_{0,n}]+I_{\mu,\lambda} | = | E_{2,\lambda}[\xi_{0,n}]-E_{2,\lambda}[\omega_n] | \rightarrow 0 \text{ as } n \rightarrow \infty\] Thus \(\{\xi_{0,n}\}\) be a minimizing sequence s.t. \(\xi_{0,n}\in K_{\mu_n}, \mu_n\rightarrow\mu\) and \(-E_{2,\lambda}[\xi_{0,n}] \rightarrow I_{\mu,\lambda}\) as \(n\rightarrow\infty\).
Knowing that \(\xi\) conserved on particle trajectory map, then \(\|\xi_n\|_{2}=\|\xi_{0,n}\|_{2}\) and \(\|x_2\xi_n\|_1=\|x_2\xi_{0,n}\|_1\). Meanwhile, since the kinetic energy is conserved in time, then \(E_{2,\lambda}[\xi_{0,n}]=E_{2,\lambda}[\xi_{n}]\) for any \(n\).
Hence \(\{\xi_{n}\}\) be a minimizing sequence s.t. \(\xi_{n}\in K_{\mu_n}, \mu_n\rightarrow\mu\) and \(-E_{2,\lambda}[\xi_{n}] \rightarrow I_{\mu,\lambda}\) as \(n\rightarrow\infty\). By the Theorem 9, by choosing a subsequence still denoted by \(\{\xi_{n}\}\), there exists \(\xi\in S_{\mu,\lambda}\) s.t. \(\omega_n\rightarrow\omega\) and \(x_2\omega_n\rightarrow x_2\omega\) in \(L^2(\Omega)\) and \(L^1(\Omega)\) respectively. Sending \(n\rightarrow\infty\), \[\begin{align} 0=&\inf_{\omega\in S_{\mu,\lambda}}\Big\{ \|\xi-\omega\|_{2}+\|x_2(\xi-\omega)\|_{1} \Big\},\\ =&\inf_{\omega\in S_{\mu,\lambda}}\Big\{ \lim_{n\rightarrow\infty}\Big( \|\xi_{n}-\omega\|_{2}+\|x_2(\xi_{n}-\omega)\|_{1}\Big) \Big\},\\ \geq&\liminf_{n\rightarrow\infty}\Big(\inf_{\omega\in S_{\mu,\lambda}}\Big\{ \|\xi_{n}-\omega\|_{2}+\|x_2(\xi_{n}-\omega)\|_{1}\Big\} \Big),\\ \geq&\epsilon_0. \end{align}\] We obtained a contradiction.
In the end, we discuss some generalizations of Theorem 24, and state some open questions.
A natural question arises as to whether the stability result applies to more general domains. It would be valuable to identify optimal conditions on the domain that ensure both the weak continuity of kinetic energy and the existence of minimizers. This is particularly relevant for domains with even weaker volume conditions and small perturbations of the strip model. Examples of such domains include shapes defined by functions like \((\ln(2 + |x_1|))^{-1}\), \(1/x\), and \(e^{-x^2/2} + 1\), along with \(\partial \mathbb{R}^2_+\) as boundaries, among others. Their behavior is unpredictable due to the absence of technical tools, such as rearrangement, and the concentrated nature resulting from the decay of the domain itself.
Since the Class [def:case32Omega61D] is very general, for a free-boundary problem whose initial domain satisfies the decay assumption 5 , it is believed that within a certain time, the free-boundary domain \(\mathcal{D}_t\) will still satisfy the condition 5 . This holds particularly for the domains studied in Hu–Luo–Yao [5]. A pertinent question arises regarding whether the stability issues discussed in this paper can be extended to the free-boundary case. This concept may be further explored by referencing the approximate Biot-Savart law proposed in [5]. However, it is important to note that their study focuses on local properties, whereas stability studies necessitate a reliance on global considerations, which requires more advanced techniques.
Our results extend the work of [1] from a single vortex patch to more general domains. A natural next step is to consider configurations with multiple patches, as studied for dipole pairs in [11]. However, extending our approach to dipole pairs presents two major difficulties.
First, the structure of the minimizer set \(S_{\mu,\nu,\lambda}\) remains largely unknown for both domain classes \(\mathcal{D}\) and \(\mathcal{S}\). Consequently, a vortex evolving under the Euler dynamics might approach minimizers with different spatial structures over its lifespan, making stability analysis significantly more challenging.
Second, the method of [11] introduces an \(L^2\)-norm equality constraint, which allows one to reinterpret minimizers of the penalized energy as maximizers of the kinetic energy. Their method heavily relies on scaling invariance, a property that is absent in our domains, as discussed in Section 1.1. Hence, a completely new approach is required to handle the multiple patch case, which is considerably more challenging.
Also, in [1], the author figured out the formula of the minimizer by the moving plane method in Theorem 4.2 of [23]. Unfortunately, when studying similarly, our model can only achieve a general structure, as outlined in Theorem 6. However, due to the lack of general symmetry and the specific characteristics of the domain, such as \(\mathbb{R}^2_+\), we are unable to derive more detailed properties. It may be possible to solve for additional useful properties by fixing a specific domain. For instance, we could explore a minimizer with compact supports (for weak finite volume domains), connected supports, boundedness, symmetry, and even an explicit formula. If we can achieve these results within this domain, we believe it may be possible to eliminate the \(L^1\) upper bound in Theorem 24, similar to what was demonstrated in [24].