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$BV$-Estimates for Non-Linear Parabolic PDE with Linear Drift

El Mahdi Erraji, Noureddine Igbida, Fahd Karami, Driss Meskine

TL;DR

The paper addresses space BV-regularity for a nonlinear parabolic PDE with a linear drift, modeled by $\frac{\partial u}{\partial t}-\Delta p+\nabla\cdot(uV)= f\in \beta(p)$ with $u\in\beta(p)$. An implicit Euler time discretization and analysis of a related stationary problem yield a weak solution $(u,p)$ and BV-control for the evolving density. Under stronger regularity assumptions on the drift and forcing, it proves local BV-estimates for $u$, and it shows a BV result for the stationary problem via a coercive, monotone framework; in the special case $\beta^{-1}\equiv 0$, the framework recovers BV for the linear transport equation. Together, these results provide compactness tools for stability and asymptotic analysis in drift-diffusion models on bounded domains, and offer a boundary-aware BV theory for nonlinear diffusion-drift PDEs.

Abstract

In the present work, we establish space Bounded Variation $(BV)$ regularity of the solution for a non-linear parabolic partial differential equations involving a linear drift term. We study the problem in a bounded domain with mixed Dirichlet-Neumann boundary conditions, a general non-linearity and reasonable assumptions on the data. Our results also cover, as a particular case, the linear transport equation in a bounded domain with an outward-pointing drift vector field on the boundary.

$BV$-Estimates for Non-Linear Parabolic PDE with Linear Drift

TL;DR

The paper addresses space BV-regularity for a nonlinear parabolic PDE with a linear drift, modeled by with . An implicit Euler time discretization and analysis of a related stationary problem yield a weak solution and BV-control for the evolving density. Under stronger regularity assumptions on the drift and forcing, it proves local BV-estimates for , and it shows a BV result for the stationary problem via a coercive, monotone framework; in the special case , the framework recovers BV for the linear transport equation. Together, these results provide compactness tools for stability and asymptotic analysis in drift-diffusion models on bounded domains, and offer a boundary-aware BV theory for nonlinear diffusion-drift PDEs.

Abstract

In the present work, we establish space Bounded Variation regularity of the solution for a non-linear parabolic partial differential equations involving a linear drift term. We study the problem in a bounded domain with mixed Dirichlet-Neumann boundary conditions, a general non-linearity and reasonable assumptions on the data. Our results also cover, as a particular case, the linear transport equation in a bounded domain with an outward-pointing drift vector field on the boundary.
Paper Structure (4 sections, 11 theorems, 96 equations)

This paper contains 4 sections, 11 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Stationary problem
  4. Proofs for the evolution problem

Key Result

Proposition 2.1

Under the assumptions $(T1)$ and $(T2),$ for any $u_0\in L^2(\Omega)$ and $f\in L^2(Q),$ the sequence of couple $(u_i,p_i)$ solving the problems sti, for $i={0,\ldots,n-1},$ is well defined ; in the sense that $u_i \in L^1(\Omega),$$p_i\in H^1_{\Gamma_D}(\Omega)$ and for any $\xi\in H^1_{\Gamma_D}(\Omega).$

Theorems & Definitions (27)

  • Proposition 2.1
  • Theorem 2.1
  • Remark 1
  • Remark 2
  • Theorem 2.2
  • Remark 3
  • Theorem 2.3
  • Remark 4
  • Proposition 3.2
  • Lemma 3.1
  • ...and 17 more