Nonlocal-to-local limit for linear transport equations with measure initial data

Technical Summary: Nonlocal-to-Local Limit for Linear Transport Equations with Measure Initial Data

Problem Statement
This work addresses the rigorous derivation of the local linear transport equation from its associated nonlocal counterpart. Specifically, the paper investigates the limit as $\varepsilon \to 0$ of the Cauchy problem:
$$\begin{cases} \partial_t u_\varepsilon + B \cdot \nabla_x (\eta_\varepsilon * u_\varepsilon) = 0, & (t, x) \in (0, \infty) \times \mathbb{R}^d \\ u_\varepsilon(0, x) = u_0, & x \in \mathbb{R}^d \end{cases}$$
where $\eta_\varepsilon(x) = \varepsilon^{-d}\eta(x/\varepsilon)$ is an approximation of the identity, $B$ is a given vector field, and the initial data $u_0$ is a finite Radon measure. The goal is to show that $u_\varepsilon$ converges to the unique distributional solution $u$ of the local equation $\partial_t u + B \cdot \nabla_x u = 0$.

While the nonlocal-to-local limit is well-studied for nonlinear conservation laws (e.g., Burgers-type equations in traffic flow models), the literature for linear transport equations is limited. Existing results often rely on one-dimensional arguments, require specific symmetry or anisotropy assumptions on the kernel $\eta$, or restrict initial data to have a distinguished sign (non-negative). This paper aims to generalize these results to arbitrary dimensions, general kernels, and signed measure initial data.

Methodology
The paper employs a multi-faceted approach combining well-posedness theory, Fourier analysis, and compactness arguments:

1. Well-Posedness Theory (Section 2):

   • The authors first establish a rigorous existence and uniqueness theory for distributional solutions to the nonlocal equation (1.1) with measure initial data.
   • For bounded vector fields $B$, they utilize a fixed-point argument (Banach contraction principle) on the integral formulation of the equation, followed by an approximation argument using mollifiers to handle $BV$ kernels.
   • For linear growth vector fields (specifically the free transport/Liouville case where $B(x,v) = (v,0)$), the standard boundedness arguments fail. Instead, the authors transform the equation into Fourier space. This converts the nonlocal PDE into a linear transport equation with a source term in the frequency domain. They apply a generalized Duhamel principle to establish existence and derive energy estimates in weighted $L^2$ spaces ($H^{-s}$).

2. Fourier Analysis and Convergence (Sections 3 & 4):

   • Constant and Free Transport: For constant $B$ and the free transport case, the authors leverage the explicit structure of the Fourier-transformed equations. They derive exact formulas for the Fourier transforms of the solutions ($\hat{u}_\varepsilon$ and $\hat{u}$). The convergence proof relies on the dominated convergence theorem applied to the difference of these Fourier representations in the $H^{-s}$ norm.
   • General Non-Positive $B$: For arbitrary non-positive $B$ in one dimension, the Fourier approach is not directly applicable due to the lack of explicit solutions. Instead, the authors utilize the specific anisotropic structure of the kernel (supported on $\mathbb{R}^-$ and non-decreasing) to prove the preservation of positivity for the solution. This allows them to derive $L^1$ bounds and use weak compactness (Banach-Alaoglu) to extract a convergent subsequence. They then identify the limit by testing against smooth functions and utilizing the properties of the odd and even parts of the kernel.

Key Contributions and Results
The paper presents three main theorems that extend the scope of nonlocal-to-local limits for linear transport equations:

• Theorem 1.1 (Constant Transport): Establishes the limit for constant vector fields $B$ in arbitrary dimensions ($d \geq 1$). The result holds for initial data in $H^{-s} \cap \mathcal{M}$ (signed measures) and covers a broad class of kernels, including those without symmetry, provided their Fourier transform satisfies specific non-degeneracy conditions (e.g., real-valued or non-vanishing real part).
• Theorem 1.2 (Free Transport/Liouville): Extends the limit to the free transport equation ($B=(v,0)$) in arbitrary dimensions. This requires the kernel $\eta$ to be even (ensuring a real-valued Fourier transform) and the initial data to be a measure with a non-vanishing Fourier transform. The convergence is proven in the $H^{-s}$ norm.
• Theorem 1.3 (Arbitrary Non-Positive $B$ in 1D): Proves the limit for general non-positive vector fields $B(x) \leq 0$ in one dimension. This result is notable for handling anisotropic kernels (supported on $\mathbb{R}^-$) and non-negative measure initial data. The proof relies on the preservation of positivity and $L^1$ estimates.

Significance and Claims
The authors claim that their work significantly generalizes the existing literature on nonlocal-to-local limits for linear transport equations in four specific ways:

1. Dimensionality: Unlike previous works that rely on genuinely one-dimensional arguments, these results are valid for arbitrary dimensions (Theorems 1.1 and 1.2).
2. Initial Data: The derivations apply to distributional solutions with measure initial data, rather than restricting to $L^p$ or smooth functions.
3. Kernel Generality: The results cover a wider class of approximations of the identity. Theorem 1.1 does not require symmetry or anisotropy, contrasting with works that impose such constraints.
4. Signed Data: The assumption that initial data must have a distinguished sign (often crucial for nonlinear Burgers-type limits) is redundant for the constant and free transport cases (Theorems 1.1 and 1.2), allowing for signed measures.

The paper concludes that while the nonlocal-to-local limit for linear transport equations was previously an open problem in higher dimensions without viscosity, the combination of Fourier methods (for linear $B$) and positivity-preserving arguments (for general non-positive $B$) provides a robust framework for these limits under significantly relaxed assumptions.