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On the $L_1$--stability for parabolic equations with a supercritical drift term

Mikhail Glazkov, Timofey Shilkin

Abstract

In this paper we investigate the existence, uniqueness and stability of weak solutions of the initial boundary value problem with the Dirichlet boundary conditions for a parabolic equation with a drift $b\in L_2$. We prove $L_1$-stability of solutions with respect to perturbations of the drift $b$ in $L_2$ in the case if the drift satisfies the ``non-spectral'' condition $\operatorname{div} b\le 0$.

On the $L_1$--stability for parabolic equations with a supercritical drift term

Abstract

In this paper we investigate the existence, uniqueness and stability of weak solutions of the initial boundary value problem with the Dirichlet boundary conditions for a parabolic equation with a drift . We prove -stability of solutions with respect to perturbations of the drift in in the case if the drift satisfies the ``non-spectral'' condition .

Paper Structure

This paper contains 6 sections, 15 theorems, 142 equations.

Table of Contents

  1. Introduction and Main Results
  2. Auxiliary results
  3. Existence and boundedness of weak solutions
  4. Proof of Theorems \ref{['Main_Theorem']} and \ref{['Stability_Theorem']}
  5. Proof of Theorem \ref{['Maximum_Princip_Theorem']}
  6. Counterexamples

Key Result

Theorem 1.1

Assume $b$, $u_0$, $f$ satisfy Problem_Data, RHS_assumption, Non-spectral. Then there exists at least one weak solution to the problem Equation which satisfies the energy estimate with some constant $c>0$ depending only on $n$, $\Omega$ and $T$.

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.2
  • Proposition 2.1
  • Proposition 2.2
  • ...and 14 more