Local Well-Posedness for Vlasov--Poisson with $L^{d+}$ Initial Density and Fractional Velocity Regularity

Technical Summary: Local Well-Posedness for Vlasov–Poisson with $L^d_+$ Initial Density and Fractional Velocity Regularity

Problem Statement
The paper addresses the local well-posedness of the Vlasov–Poisson system in dimensions $d \ge 2$:
$$\partial_t f + v \cdot \nabla_x f + E \cdot \nabla_v f = 0, \quad E = \sigma \nabla_x \Delta^{-1} \rho, \quad \rho(t, x) = \int_{\mathbb{R}^d} f(t, x, v) \, dv.$$
The central difficulty lies in the anisotropic nature of the problem: the distribution function $f$ lives on phase space, while the self-consistent electric field $E$ is generated solely by the spatial density $\rho$, which is a velocity average of $f$. Classical theories often require strong regularity on $f$ in the full phase space (e.g., Sobolev spaces $H^s(\mathbb{R}^{2d})$) or uniform boundedness of the density $\rho \in L^\infty_x$. This work investigates a regime where the initial datum $f_0$ possesses no positive-order spatial derivatives, but instead satisfies an anisotropic Hölder condition in the velocity variable with a spatial $L^p$ control, where $p > d$. The goal is to establish local existence, uniqueness, and continuous dependence for initial data that may be unbounded in space ($\rho_0 \notin L^\infty_x$) but possess sufficient velocity regularity to induce spatial regularity through transport.

Methodology
The proof relies on a density-based fixed-point argument combined with a characteristic reparametrization technique. The methodology proceeds through four main steps:

1. Characteristic Reparametrization:
   Instead of solving the transport equation directly, the author studies the backward characteristic flow $(X_{s,t}, V_{s,t})$ generated by a prescribed electric field $E$. A key innovation is the construction of a global diffeomorphism that maps the terminal velocity to the initial position. Specifically, for a fixed time $t$ and position $x$, the map $v \mapsto X_{0,t}(x, v)$ is inverted to express the initial position $y$ and initial velocity $w$ in terms of $x$ and $v$. This yields a "nonlinear mixing" formula for the density:
   $$\rho(t, x) = t^{-d} \int_{\mathbb{R}^d} f_0\left(y, W_t(x, y)\right) J_t(x, y) \, dy,$$
   where $W_t(x, y)$ is the initial velocity corresponding to a particle arriving at $x$ at time $t$ starting from $y$, and $J_t$ is the associated Jacobian. This formula generalizes the free-transport relation $\rho_{free}(t, x) = t^{-d} \int f_0(y, (x-y)/t) dy$.

2. Dispersive Estimates and Regularity Transfer:
   The core estimate demonstrates that the fractional Hölder regularity in velocity ($C^\theta_v$) is converted into spatial Hölder regularity ($C^\alpha_x$) via the ballistic mixing inherent in the transport.

   • The assumption is that $f_0$ has a weighted uniform $C^\theta_v$ modulus $G_\theta(x)$ belonging to $L^p_x$.
   • The characteristic map $W_t(x, y)$ behaves like $(x-y)/t$ plus a perturbation controlled by the electric field.
   • By analyzing the increment $x \mapsto x+\ell$, the author shows that the $C^\theta_v$ regularity of $f_0$ translates to a $C^\alpha_x$ regularity of $\rho(t)$ with a time-singular bound:
     $$[\rho(t)]_{C^\alpha_x} \lesssim t^{-d/p - \alpha} (\|G_0\|_{L^p} + \|G_\theta\|_{L^p}).$$
   • Crucially, because $p > d$ and $\alpha$ is chosen such that $d/p + \alpha < 1$, the singularity $t^{-d/p-\alpha}$ is integrable in time. This ensures the electric field $E = \nabla \Delta^{-1} \rho$ belongs to $L^1(0, T; C^{1, \alpha}_x)$, which is sufficient to close the characteristic construction.

3. Schauder Fixed Point:
   Since the map from density to density is not a contraction in the relevant norms, the Banach fixed-point theorem is not applicable. Instead, the author constructs a closed, convex set of admissible densities in the Banach space $X = L^r(0, T; C^\beta(\mathbb{R}^d))$.

   • Compactness: The compactness of the solution map is established using the Aubin–Lions–Simon theorem. This relies on the dispersive bounds (providing spatial regularity), the continuity equation (providing time regularity of the density in negative Sobolev spaces), and spatial tightness (derived from the finite mass and velocity decay of $f_0$).
   • Existence: The Schauder fixed-point theorem guarantees the existence of a fixed point $\rho$, which defines a Lagrangian solution $(f, E)$.

4. Uniqueness via Loeper-type Stability:
   Uniqueness is proven using a stability estimate inspired by Loeper [26]. The standard Loeper estimate requires bounded density. Here, the density $\rho(t)$ is unbounded near $t=0$ (behaving like $t^{-d/p}$). However, since $d/p < 1$, the $L^\infty$ norm is integrable in time.

   • The author defines a quantity $P(t)$ measuring the distance between two flows in phase space.
   • Using the Loeper estimate pointwise in time and integrating the resulting differential inequality via Gronwall's lemma, uniqueness is established. The integrability of the singularity allows the application of the stability estimate at almost every time $t > 0$.

Key Contributions and Results

• Theorem 1 (Local Well-Posedness): For $d \ge 2$, $p > d$, and $0 < \theta \le 1$, if the initial datum $f_0$ satisfies finite mass, weighted velocity supremum $G_0 \in L^p_x$, and weighted velocity Hölder modulus $G_\theta \in L^p_x$, then there exists a unique Lagrangian solution on a short time interval $[0, T]$.
• Regularity of the Solution: The solution satisfies:
  • $f \in C([0, T]; L^1 \cap L^p(\mathbb{R}^{2d}))$.
  • $\rho \in L^\infty(0, T; L^1 \cap L^p) \cap L^r(0, T; C^\alpha)$.
  • $E \in L^r(0, T; C^{1, \alpha})$.
  • The density exhibits a time-singular $L^\infty$ bound: $\|\rho(t)\|_{L^\infty} \lesssim t^{-d/p}$.
• Anisotropic Assumptions: The result requires no spatial derivatives of $f_0$. The regularity is purely anisotropic: fractional Hölder continuity in velocity, controlled in $L^p$ spatially. This distinguishes the result from previous Sobolev-based theories (e.g., Jeong–Tae [24], Tae [34]) which require full phase-space regularity.
• Uniqueness in a New Class: The paper establishes uniqueness for initial data where the density is not uniformly bounded in time, extending the classical Loeper uniqueness regime which relies on $\rho \in L^\infty_{t,x}$.

Significance and Claims
The paper claims to provide a local well-posedness criterion that operates beyond the classical regime of uniformly bounded densities.

• Mechanism: The primary mechanism is the "ballistic mixing" of the transport equation, which converts velocity regularity into spatial regularity. This allows the electric field to gain enough regularity ($C^{1, \alpha}$) to define a stable flow, even when the initial density is singular.
• Sharpness of Conditions: The authors note that the conditions $p > d$ and $\theta > 0$ are sharp for the specific Hölder-flow argument used.
  • If $p = d$, the $L^\infty$ singularity becomes $t^{-1}$, which is not integrable, breaking the time-integrability of the field regularity.
  • If $\theta = 0$, there is no mechanism to generate spatial Hölder regularity from the initial data, even for free transport.
• Open Problems: The paper explicitly states that global well-posedness under these assumptions remains open. Furthermore, it poses the question of whether non-uniqueness occurs for data with $p > d$ but $\theta = 0$ (i.e., only weighted $L^p$ control without velocity regularity), suggesting that the current result does not cover the endpoint $\theta = 0$. It also highlights that while non-uniqueness is known for 2D Euler in certain weak classes, the kinetic structure of Vlasov–Poisson (positivity, elliptic constraint) may prevent similar pathologies, leaving the uniqueness question for "barely" unbounded densities as a fundamental open problem.

In summary, the paper establishes that for the Vlasov–Poisson system, fractional velocity regularity is sufficient to compensate for a lack of spatial regularity and spatial boundedness, provided the spatial integrability exponent $p$ is super-critical ($p > d$). This is achieved through a novel combination of characteristic reparametrization, dispersive estimates, and a density-based fixed-point argument.