March 23, 2026
A Moser-type construction for the kinetic Liouville equation is proposed, which is based on a characteristic-adapted interpolation. For Hamiltonian accelerations, the construction is reduced to a family of weighted elliptic problems in the velocity variable. The corresponding kinetic compatibility condition is derived.
In the seminal work [1], J. Moser shows that on a compact manifold without boundary, any pair of smooth and positive volume forms of equal total mass can be connected by a diffeomorphism. Later developments of this result, in particular [2], provide a constructive approach based on interpolating the densities and solving an elliptic Neumann problem for a time-dependent vector field whose flow realizes the desired transport.
More precisely, let \(\Omega\subset \mathbb{R}^n\) be a bounded connected domain with smooth, i.e. \(C^\infty\) boundary, and let \(f,g\in C^\infty(\overline{\Omega})\) satisfy \(f>0\), \(g>0\) on \(\overline{\Omega}\) and be such that \[\int_\Omega f(x)\,dx=\int_\Omega g(x)\,dx.\] Then [1] states the existence of a diffeomorphism \(\Gamma: \overline{\Omega} \to \overline{\Omega}\) pointwise fixing the boundary, such that \(\Gamma_\#(f\,dx)=g\,dx\), namely \[f(x)=g(\Gamma(x))\det D\Gamma(x), \quadfor everyx \in \Omega.\] To construct such a diffeomorphism, a possible strategy, following the notation of Dacorogna and Moser [2] (see also [3]), is to connect the initial and final densities by a smooth path, for instance, the linear interpolation \[\rho(x,t)=(1-t) \, f(x)+t \, g(x) , \qquad t\in[0,1], \, x \in \Omega . \label{origInterpolation}\tag{1}\] The goal is then to find a time-dependent velocity field \(b(x,t)\) generating this path as a solution of the continuity equation \[\label{eq:cont} \partial_t \rho(x,t) + \nabla \cdot \big(b(x,t) \, \rho(x,t)\big) = 0, \qquad \rho(\cdot,0)=f,\tag{2}\] so that \(\rho(\cdot,1)=g\). Hence, denoting by \(\Gamma_t\) the flow associated with the vector field \(b\), the solution of the continuity equation 2 can be written as \(\rho(\cdot,t)dx=(\Gamma_t)_{\#}(f\,dx)\). In particular, the \(1\)-time flow \(\Gamma_1\) pushes forward the initial density onto the final one \[(\Gamma_1)_\# (f\,dx)=g\,dx ,\] providing the diffeomorphism solving the prescribed transport problem.
Differentiating 1 , \(\partial_t\rho=g-f.\) Therefore, the problem is to construct a flux \(w:= b \, \rho\) satisfying \(\nabla\cdot w=f-g\). This goal can be achieved setting \(w=-\nabla\phi\) and solving the elliptic problem \[\label{eq:moser95poisson} \begin{cases} -\Delta \phi=f - g & \text{in }\Omega,\\ \partial_n\phi=0 & \text{on }\partial\Omega, \end{cases}\tag{3}\] (or an appropriate decay condition as \(x \to \infty\)) together with the normalization \[\int_{\Omega}\phi(x)\,dx=0.\]
In the present work, we investigate the extension of this construction to the case of phase-space densities of Hamiltonian systems. In this setting, the continuity equation governing the evolution of densities takes the form of \[\label{eqLiouville} \partial_t \rho(x,v,t) + v\cdot\nabla_x \rho(x,v,t) + \nabla_v \cdot \big( a(x,v,t) \, \rho(x,v,t) \big)= 0,\tag{4}\] where \((x,v)\in \Omega\subset {\mathbb{R}}^n\times{\mathbb{R}}^n\). More precisely, the problem is formulated as follows.
Let \(\Omega = X\times V\), with \(X={\mathbb{R}}^n\) or \({\mathbb{T}}^n\) (\(n\)-dimensional torus) and \(V = {\mathbb{R}}^n\). Let \(f,g\in C^\infty(\Omega)\) be strictly positive and such that \[\int_\Omega f(x,v) \,dx\,dv = \int_\Omega g(x,v) \,dx\,dv.\] Consider the Liouville equation 4 with initial condition \(\rho(0)=f\).
Find \(a:\Omega \times (0,1) \to {\mathbb{R}}^n\) such that the associated solution \(\rho(t)\) satisfies \(\rho(1)=g\).
This problem can be intepreted as a controllability problem for the Liouville equation 4 .
Equation 4 , called Liouville equation by L. Boltzmann in his lectures [4], plays a fundamental role in kinetic theory and in the statistical description of non-interacting many-particle systems, where \(\rho\) denotes the normalized material density function. At the microscopic level, our Liouville equation governs the evolution of the phase-space density of the particles’ dynamics given by the differential system \[\label{eq:micro} \begin{cases} \chi'(t) &=\upsilon(t), \\ \upsilon'(t) &=a(\chi(t),\upsilon(t),t), \end{cases}\tag{5}\] where the spatial drift is fixed to be the velocity variable \(\upsilon\), while the force (acceleration) field constitutes the only degree of freedom.
Moser’s approach has been studied from various perspectives in the context of the continuity equation also within the framework of control theory [5]–[7]. To the best of our knowledge, the corresponding problem for the Liouville equation 4 , defined on the phase space, is an open problem. It requires to resolve the significant constraint due to the fact that the drift in the \(x\)-direction is prescribed and equal to \(v\), which is an independent variable.
Our discussion can be placed at the intersection of several classical theories. The original inspiration comes from the volume-form construction of J. Moser [1] and its constructive reformulation by B. Dacorogna and J. Moser [2]. In that framework, the transport field is essentially arbitrary and the only condition on the initial and final densities is the equality of total mass. The Liouville setting differs because the admissible phase-space velocity fields are constrained by the kinetic structure\((\dot{x},\dot{v})=(v,a).\)
In particular, in what follows we will focus on the conservative Hamiltonian case in which \[(\dot{x},\dot{v})=(v,-\nabla_x V).\] The conservative Hamiltonian case is related to the classical Liouville theorem in Hamiltonian mechanics, already present in the works of J. Liouville and later systematically developed in statistical mechanics and Hamiltonian dynamics. In this case, Hamiltonian flows preserve phase-space volume and transport densities by measure-preserving rearrangements: \[g=f\circ\Psi^{-1},\] where \(\Psi\) is the Hamiltonian flow map. Consequently, all rearrangement invariants are preserved. In this framework, a equimeasurability condition arises.
The distinction between arbitrary volume-preserving maps and Hamiltonian flows is closely connected with the rigidity phenomena of symplectic geometry initiated by Gromov through the non-squeezing theorem [8] and further developed in modern symplectic topology; see for instance [9], [10]. In this framework, Hamiltonian diffeomorphisms constitute a much more rigid class than general measure-preserving maps, since they preserve not only phase-space volume but also the underlying symplectic structure.
The preservation of quantities of the form \(\int_{\Omega}\Phi(f)\) is also related to the theory of Casimir invariants in kinetic and fluid equations. Such invariants play a fundamental role in the analysis of the Vlasov equation, incompressible Euler equations, and related transport problems; see for example [11], [12]. Rearrangement methods and measure-preserving transport techniques have also been extensively developed in optimal transport and fluid mechanics, particularly in the works of Y. Brenier [13]–[15].
The present discussion suggests a hierarchy of compatibility conditions for endpoint reachability problems associated with the Liouville equation:
for arbitrary accelerations \(a(x,v,t)\), one may expect that only conservation of total mass is required;
for time-dependent conservative Hamiltonian accelerations \(a=-\nabla_x V(x,t)\), equimeasurability becomes a necessary condition; (see Proposition 1 below)
for autonomous Hamiltonians \(a=-\nabla_x V(x)\) the full Hamiltonian energy distribution must be preserved; (see Proposition 2 below)
for general integrable Hamiltonian systems, further invariants related to action variables and invariant tori may appear.
We consider the case in which the acceleration is generated by a time-varying potential: \[a(x,t)=-\nabla_x V(x,t).\] Then the phase-space dynamics is a Hamiltonian system as follows \[\label{eq:hamtimedep} \begin{cases} \dot{x} &=v,\\ \dot{v} &=-\nabla_x V(x,t), \end{cases}\tag{6}\] corresponding to the Hamiltonian \[H(x,v,t)=\frac{1}{2} |v|^2+V(x,t).\] In this case a necessary condition for the reachability of \(g\) from \(f\) via the flow of 6 is equimisurability as stated in the following proposition.
Proposition 1. Let \(\rho(x,t)\) be the solution of the Liouville equation \[\partial_t\rho + v\cdot\nabla_x\rho - \nabla_x V(x,t)\cdot\nabla_v\rho = 0, \qquad \rho(\cdot,0)=f,\] such that \[\rho(\cdot,1)=g .\] Then \(f\) and \(g\) are equimeasurable, namely \[\int_{\Omega} \Phi(f(x,v))\,dx\,dv = \int_{\Omega} \Phi(g(x,v))\,dx\,dv\] for any measurable function \(\Phi:{\mathbb{R}}\to {\mathbb{R}}\).
Proof. The phase-space vector field of 6 is divergence-free since \(\nabla_x\cdot v+\nabla_v\cdot(-\nabla_x V)=0\). As a consequence, the associated flow is volume-preserving in phase space. If \(\Psi_t\) denotes the Hamiltonian flow, then the solution of 4 is \[\rho(t)=\rho(0)\circ \Psi_t^{-1}.\] The Hamiltonian flow \(\Psi_t\) preserves phase-space volume (Liouville theorem), that is, \[\det D\Psi_t(x,v)=1 .\] Along characteristics, the density satisfies \[\frac{d}{dt}\rho(\Psi_t(x,v),t)=0,\] hence \(\rho(\Psi_t(x,v),t)=f(x,v)\). At time \(t=1\) we obtain \(g(x,v)=f(\Psi_1^{-1}(x,v))\).
Fix \(\lambda\in{\mathbb{R}}\) and define the level sets \[E_f(\lambda) = \{z\in\Omega:f(z)>\lambda\}, \qquad E_g(\lambda) = \{z\in\Omega:g(z)>\lambda\}.\] Since \(g(z)=f(\Psi_1^{-1}(z))\), we have \[z\in E_g(\lambda) \iff \Psi_1^{-1}(z)\in E_f(\lambda),\] that is, \[E_g(\lambda)=\Psi_1(E_f(\lambda)).\]
Because \(\Psi_1\) is volume-preserving, \[|E_g(\lambda)| = |\Psi_1(E_f(\lambda))| = |E_f(\lambda)|.\] Therefore \[|\{g>\lambda\}| = |\{f>\lambda\}|,\] and the conclusion follows. ◻
Equimeasurability implies, in particular, that \[\int_{\Omega} f^p\,dx\,dv = \int_{\Omega} g^p\,dx\,dv, \qquad 1\le p<\infty .\] Thus, in the Hamiltonian setting, equality of total mass is not sufficient. The Hamiltonian flow preserves the full distribution of density values.
Remark 1. Equimeasurability is only a necessary condition for reachability by a Hamiltonian force. Indeed, if \[g=f\circ\Psi_1^{-1},\] where \(\Psi_t\) is the Hamiltonian flow generated by \(H(x,v,t)\), then \(f\) and \(g\) are equimeasurable because \(\Psi_1\) is volume-preserving.
The converse, however, is not true in general. Equimeasurability only implies that there exists some volume-preserving map \(T\) such that \[g=f\circ T^{-1},\] under appropriate regularity assumptions. It does not imply that \(T\) is the time-one map of a Hamiltonian system of the mechanical form \[\dot{x}=v,\qquad \dot{v}=-\nabla_x V(x,t).\] This class is much smaller than the class of all volume-preserving maps.
Thus, the construction of \(V\) requires additional geometric compatibility, that is, the desired rearrangement must be realizable by a mechanical Hamiltonian flow.
This is an overdetermined constraint because the unknown force \(-\nabla_x V(x,t)\) depends only on \(x\) and \(t\), not on \(v\). Therefore, even when \(f\) and \(g\) are equimeasurable, a conservative potential \(V\) may not exist.
If the potential is independent of time, namely \[V=V(x),\] then the Hamiltonian \[H(x,v)=\frac{1}{2} |v|^2+V(x)\] is conserved along characteristics \[\frac{d}{dt}H(x(t),v(t))=0.\] The Hamiltonian flow cannot move mass between different energy levels. If \(g\) is reachable from \(f\) by the autonomous Hamiltonian flow, then, the energy marginal distributions of \(f\) and \(g\) must necessarily agree: \[\int_{\{H\le E\}} f(x,v)\,dx\,dv = \int_{\{H\le E\}} g(x,v)\,dx\,dv \qquad \text{for all }E .\]
Proposition 2. Let \(\rho(x,t)\) be the solution of the Liouville equation \[\partial_t\rho + v\cdot\nabla_x\rho - \nabla_x V(x)\cdot\nabla_v\rho = 0, \qquad \rho(\cdot,0)=f,\] such that \[\rho(\cdot,1)=g.\] Let \[H(x,v)=\frac{1}{2} |v|^2+V(x)\] then for every measurable function \(\Phi : {\mathbb{R}}\to {\mathbb{R}}\), it must hold \[\int_{\Omega} \Phi(H(x,v))\, f(x,v)\,dx\,dv = \int_{\Omega} \Phi(H(x,v))\, g(x,v)\,dx\,dv .\]
Thus, in the autonomous case, equimeasurability is not sufficient. Besides preserving phase-space volume and all \(L^p\)-norms of the density, the flow also preserves the distribution of mass over the invariant energy surfaces \[H(x,v)=E .\] Consequently, a time-independent conservative potential can only rearrange density along its own Hamiltonian trajectories and cannot perform arbitrary equimeasurable rearrangements of phase space.
In order to adapt the Dacorogna-Moser constructive proof to the kinetic Liouville structure we consider the characteristics lines associated \[(x(t),v(t))=(x_0+t v_0,v_0) ,\] for chosen initial conditions \(x_0\) and \(v_0\), associated the free transport equation \[\partial_t\rho+v\cdot\nabla_x\rho=0,\] corresponding to the case when \(a=0\) in the Liouville equation. Hence the free evolution of a density \(f\) is given by \[\rho(x,v,t)=f(x-tv,v).\] Instead of the standard Moser interpolation we consider a characteristic-adapted interpolation of the form \[\rho_t(x,v) = (1-t) \, f(x-tv,v) + t \, g(x+(1-t)v,v). \label{newInterpolation}\tag{7}\] incorporating the horizontal kinetic drift generated by the free characteristics.
Notice that the endpoint conditions are satisfied: \(\rho_0=f\) and \(\rho_1=g\). We also have \[\begin{align} \partial_t \rho_t(x,v) & = - f(x-tv,v) + g(x+(1-t)v,v) - (1-t) \, v \, \partial_x f(x-tv,v) \\ & \qquad - t \, v \, \partial_x g(x+(1-t)v,v) \\ & = - f(x-tv,v) + g(x+(1-t)v,v) - v \cdot \partial_x \rho_t (x,v) . \end{align}\] This calculation provides the defect \[\label{defect} \partial_t\rho_t+v\cdot\nabla_x\rho_t = g(x+(1-t)v,v) - f(x-tv,v) .\tag{8}\]
Our approach is then to assume that the admissible vertical \(v\)-flux \(q_t\) is generated by a potential in the velocity variable: \[q_t=\rho_t\nabla_v U_t .\] Then the Liouville equation \[\partial_t\rho_t + v\cdot\nabla_x\rho_t + \nabla_v\cdot q_t = 0\] becomes \[\nabla_v\cdot(\rho_t\nabla_v U_t) = - \big( \partial_t\rho_t + v\cdot\nabla_x\rho_t \big). \label{ellipticEqnew}\tag{9}\] This is a weighted elliptic equation in the velocity variable parametrized by \((x,t)\) and its solvability will be discussed later in this section. Notice that these problems are uniformly elliptic only if the interpolation density \(\rho_t\) remains strictly positive.
In this setting, the associated acceleration field is given by \[a_t=\nabla_v U_t\] and at the level of particles, the dynamics is given by the differential system \[\dot{x}=v, \qquad \dot{v}=\nabla_v U_t(x,v).\]
Remark 2. Assuming homogeneous Neumann boundary conditions in \(v\) or sufficiently fast decay conditions, the solvability condition for the elliptic equation 9 is given by \[\label{solvcond} \int_V \big( \partial_t\rho_t + v\cdot\nabla_x\rho_t \big)\,dv = 0 ,\qquad{(1)}\] for every fixed \(x\) and \(t\).
Thus the interpolation path \(\rho_t\) must satisfy the following kinetic compatibility condition \[\partial_t\int_V\rho_t\,dv + \nabla_x\cdot\int_V v\rho_t\,dv = 0 .\] If we introduce the spatial density \[m_t(x)=\int_V\rho_t(x,v)\,dv\] and the momentum density \[j_t(x)=\int_V v\rho_t(x,v)\,dv ,\] the above condition becomes the macroscopic continuity equation \[\partial_t m_t+\nabla_x\cdot j_t=0 .\]
Remark 3. Because of 8 and the solvability condition for the elliptic problem ?? , we have that \(f\) and \(g\) must satisfies \[\int_V f(x-tv,v)\,dv = \int_V g(x+(1-t)v,v)\,dv .\] Unlike the classical Dacorogna-Moser construction, where solvability reduces to equality of total mass, the present kinetic setting imposes a family of characteristic
marginal constraints indexed by \(t\in[0,1]\).
However we stress that this condition originates from the assumption that the admissible vertical \(v\)-flux \(q_t\) is generated by a potential in the velocity variable and from the choice
of the characteristc-adapted interpolation 7 . Therefore it is not a necessary condition for the existence of a generic suitable acceleration sending \(f\) to \(g\).
We now investigate solvability of the weighted elliptic problem in 9 .
Proposition 3. Let \(f,g\in C^\infty(\Omega)\) satisfy \(f>0\), \(g>0\) on \(\Omega\) and be such that \[\int_{{\mathbb{R}}^n} f(x-tv,v)\,dv = \int_{{\mathbb{R}}^n} g(x+(1-t)v,v)\,dv,\] for every \(x\in X,t\in [0,1]\). Fix \(x\in X,t\in [0,1]\) and consider the measure \[\label{measure} d\mu_t^x(v) = \rho_t(x,v)\,dv ,\qquad{(2)}\] where \[\rho_t(x,v) = (1-t)f(x-tv,v) + t\,g(x+(1-t)v,v),\] and the associated weighted Sobolev space \[\label{wsob} H^1(\mu_t^x) = \left\{ u :{\mathbb{R}}^n \to {\mathbb{R}}\,\,\,\,\, \mathrm{s.t.}\, \int_{{\mathbb{R}}^n} |u|^2\,d\mu_t^x + \int_{{\mathbb{R}}^n} |\nabla_v u|^2\,d\mu_t^x < \infty \right\}.\qquad{(3)}\] If, for every \(x\in X, t\in[0,1]\), there exists a constant \(C_P^{x,t}\) such that, defining \[\bar u = \frac{\int_{{\mathbb{R}}^n} u\,d\mu_t^x}{\int_{{\mathbb{R}}^n} d\mu_t^x},\] the Poincaré inequality \[\int_{{\mathbb{R}}^n} |u-\bar u|^2\,d\mu_t^x \le C_P^{x,t} \int_{{\mathbb{R}}^n} |\nabla_v u|^2\,d\mu_t^x , \label{ePFinequality}\qquad{(4)}\] holds for every \(u \in H^1(\mu_t^x)\), then there exists \(a:\Omega \times [0,1] \to {\mathbb{R}}^n\) such that the associated solution of the Liouville equation 4 is exactly \(\rho_t\). In particular, it satisfies \(\rho_0=f\) and \(\rho_1=g\).
The proof splits into two steps. In Lemma 1, we show the relation between the solution of the continuity equation, in which the acceleration \(a(x,v,t)\) is chosen to be generated by a potential \(U_t^x(v)\), and the elliptic equation that this potential must satisfy. In Lemma 2, we show that under the assumptions of the proposition, the elliptic problem admits a unique solution, modulo additive constants. Then the conclusion follows taking \(a(x,v,t)=\nabla_vU_t^x(v)\).
Lemma 1. Let \(U_t^x\) be a solution of \[\nabla_v\cdot(\rho_t\nabla_v U_t^x) = f(x-tv,v)-g(x+(1-t)v,v).\] Then \(\rho_t\) solves the Liouville equation 4 with \[a(x,v,t)=\nabla_v U_t^x(v).\] In particular, it satisfies \(\rho_0=f\) and \(\rho_1=g\).
Proof. Recall that \[\rho_t(x,v) = (1-t) \, f(x-tv,v) + t \, g(x+(1-t)v,v).\] Hence \[\begin{align} \partial_t \rho_t(x,v) & = - f(x-tv,v) + g(x+(1-t)v,v) - (1-t) \, v \, \partial_x f(x-tv,v) \\ & \qquad - t \, v \, \partial_x g(x+(1-t)v,v) \\ & = - f(x-tv,v) + g(x+(1-t)v,v) - v \cdot \nabla_x \rho_t (x,v). \end{align}\] Which gives \[\begin{align} \partial_t \rho_t(x,v) + v \cdot \nabla_x \rho_t (x,v) & = g(x+(1-t)v,v) - f(x-tv,v)\\ & = - \nabla_v\cdot(\rho_t\nabla_v U_t^x). \end{align}\] Then \[\partial_t\rho_t+v\cdot\nabla_x\rho_t + \nabla_v a_t(x,v) = 0.\] ◻
Lemma 2. Consider the measure \(d\mu_t^x(v)\) and the associated weighted Sobolev space \(H^1(\mu_t^x)\) as defined in ?? ,?? .
Consider the equation \[\label{eq:elliptic} \nabla_v\cdot(\rho_t(x,v)\nabla_v U_t^x(x,v)) = h_t(x,v),\qquad{(5)}\] where \[h_t(x,v) = f(x-tv,v)-g(x+(1-t)v,v).\] Assume that, for every \(x \in X\) and \(t\in[0,1]\), the Poincaré inequality ?? holds for every \(u \in H^1(\mu_t^x)\). Then ?? with homogeneous Neumann boundary conditions in \(v\) or sufficiently fast decay conditions has a unique solution in the space \[\left\{ u\in H^1(\mu_t^x): \int_{{\mathbb{R}}^n}u\,d\mu_t^x=0 \right\}.\]
Proof. The weak formulation of the elliptic problem ?? is \[\int_{{\mathbb{R}}^n} \nabla_v U_t^x\cdot\nabla_v\varphi \,d\mu_t^x = - \int_{{\mathbb{R}}^n} h_t\varphi\,dv, \qquad \varphi\in H^1(\mu_t^x).\] Since constants belong formally to the kernel of the operator, uniqueness can only hold modulo additive constants. Therefore for uniqueness we impose the normalization condition \[\int_{{\mathbb{R}}^n} U_t^x\,d\mu_t^x=0 . \label{eUniqCond}\tag{10}\] The Poincaré inequality ensures that the bilinear form \[B(u,\varphi) = \int_{{\mathbb{R}}^n} \nabla_v u\cdot\nabla_v\varphi \,d\mu_t^x\] is coercive on the subspace \[\left\{ u\in H^1(\mu_t^x): \int_{{\mathbb{R}}^n}u\,d\mu_t^x=0 \right\}.\] Consequently, by the Lax–Milgram theorem, one obtains existence and uniqueness of weak solutions. ◻
Example 1. The essential analytical issue becomes the validity of the weighted Poincaré inequality associated with the interpolation density \(\rho_t\). A classical sufficient condition is provided by Gaussian confinement. Assume that there exist positive constants \(c_0,c_1,C_0,c_2\) such that for any \(x \in X\) and \(t \in [0,1]\) \[c_0 e^{-c_1|v|^2} \le \rho_t(x,v) \le C_0 e^{-c_2|v|^2}, \qquad v\in{\mathbb{R}}^n.\]
Then the measure \(d\mu_t^x=\rho_t\,dv\) is uniformly comparable with a Gaussian measure. Consequently, classical weighted Poincaré inequalities for Gaussian-type measures apply; see for instance the Bakry-Émery theory [16]. Therefore there exists a constant \(C>0\), depending only on the comparison constants \(c_0,c_1,C_0,c_2\), such that ?? holds.
We now show that such Gaussian confinement estimates hold for suitable Gaussian endpoint densities. Consider the characteristic interpolation \[\rho_t(x,v) = (1-t)f(x-tv,v) + t\,g(x+(1-t)v,v).\] Assume that the endpoint densities satisfy Gaussian bounds of the form \[a_f e^{-\alpha_f|v|^2-\beta_f} \le f(x-tv,v) \le A_f e^{-\gamma_f|v|^2+B_f},\] and \[a_g e^{-\alpha_g|v|^2-\beta_g} \le g(x+(1-t)v,v) \le A_g e^{-\gamma_g|v|^2+B_g},\] for any \(x\), \(v\), and \(t\in[0,1]\).
We first derive the lower bound. If \(0\le t\le\tfrac12\), then \[\rho_t(x,v) \ge \frac{1}{2} f(x-tv,v) \ge \frac{1}{2} a_f e^{-\alpha_f|v|^2-\beta_f}.\]
If \(\tfrac12\le t\le1\), then \[\rho_t(x,v) \ge \frac{1}{2} g(x+(1-t)v,v) \ge \frac{1}{2} a_g e^{-\alpha_g|v|^2-\beta_g}.\]
Consequently, we obtain \[\rho_t(x,v) \ge c_0 e^{-c_1|v|^2},\] with \[c_0 = \frac{1}{2} \min\{a_f e^{-\beta_f},a_g e^{-\beta_g}\}, \qquad c_1 = \max\{\alpha_f,\alpha_g\}.\]
We now derive the upper bound. Since \(t \in [0,1]\), we have \[\begin{align} \rho_t(x,v) &= (1-t)f(x-tv,v) + t\,g(x+(1-t)v,v) \\ &\le f(x-tv,v) + g(x+(1-t)v,v) \\ &\le A_f e^{-\gamma_f|v|^2+B_f} + A_g e^{-\gamma_g|v|^2+B_g}. \end{align}\]
Hence we obtain \[\rho_t(x,v) \le C_0 e^{-c_2|v|^2},\] where \[c_2 = \min\{\gamma_f,\gamma_g\}, \qquad C_0 = A_f e^{B_f}+A_g e^{B_g}.\]
Therefore the interpolation density satisfies the two-sided Gaussian estimate \[c_0 e^{-c_1|v|^2} \le \rho_t(x,v) \le C_0 e^{-c_2|v|^2}.\]
In particular, the weighted elliptic problem associated with the characteristic interpolation admits a coercive variational formulation in weighted Sobolev spaces, yielding together with 10 existence and uniqueness of weak solutions.
The preceding construction shows that a Moser-type strategy for the Liouville equation is possible once an interpolation path \(\rho_t\) has been prescribed. However, unlike the classical Dacorogna-Moser construction for the continuity equation, the interpolation cannot be regarded as an auxiliary independent choice, but rather as part of the problem.
In the Liouville equation, \[\partial_t\rho + v\cdot\nabla_x\rho + \nabla_v\cdot(\rho a)=0,\] the phase-space characteristics are constrained by \[\dot{x}=v, \qquad \dot{v}=a(x,v,t).\] Thus the horizontal transport is prescribed by the velocity variable itself, and only the vertical component is free. Consequently, fixing an interpolation path already imposes a strong restriction on the admissible characteristic geometry.
After prescribing \(\rho_t\), the Liouville equation becomes \[\nabla_v\cdot(\rho_t a_t) = - \big( \partial_t\rho_t + v\cdot\nabla_x\rho_t \big).\] Under the potential ansatz \(a_t=\nabla_v U_t\), one obtains the weighted elliptic equation \[\nabla_v\cdot(\rho_t\nabla_v U_t) = - \big( \partial_t\rho_t + v\cdot\nabla_x\rho_t \big).\] Assuming homogeneous Neumann boundary conditions in \(v\) or sufficiently fast decay, its solvability condition is \[\int_V \big( \partial_t\rho_t + v\cdot\nabla_x\rho_t \big)\,dv = 0 ,\] or as we have seen \[\partial_t m_t+\nabla_x\cdot j_t=0 .\]
Therefore the compatibility condition arising from the potential ansatz is not merely a property of the endpoints \(f\) and \(g\), but of the entire interpolation path.
In particular, choosing the characteristics interpolation \[\rho_t(x,v) = (1-t)f(x-tv,v) + t g(x+(1-t)v,v),\] we have seen that the solvability condition reduces to the property of the endpoints of satisfying the equality \[\int_V f(x-tv,v)\,dv = \int_V g(x+(1-t)v,v)\,dv,\] for any \(x\in X\), \(t \in [0,1]\).
In any case, failure of the associated compatibility condition does not imply failure of Liouville reachability. It only implies failure of this particular Moser-type construction based on the potential ansatz and on this characteristic-adapted interpolation.
This suggests that a genuinely general Moser construction for the Liouville equation should treat the interpolation itself as part of the unknown.
Institut für Mathematik, Universität Würzburg, Emil-Fischer-Strasse 30, 97074 Würzburg, Germany. e-mail: alfio.borzi@mathematik.uni-wuerzburg.de↩︎
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy. e-mail: caponigro@mat.uniroma2.it↩︎
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy. e-mail: vicari@axp.mat.uniroma2.it↩︎