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<p>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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(b\in L_2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>∈</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. We prove uniform in time <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-stability of solutions with respect to perturbations of the drift <i>b</i> in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(L_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> in the case if the drift satisfies the “non-spectral” condition <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\operatorname {div} b\le 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>div</mo> <mi>b</mi> <mo>≤</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On the \({L_1}\)–stability for parabolic equations with a supercritical drift term

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  • 发表日期：

    2026年3月10日

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  • Mikhail Glazkov,
  • Timofey Shilkin

摘要

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 \(b\in L_2\) b ∈ L 2 . We prove uniform in time \(L_1\) L 1 -stability of solutions with respect to perturbations of the drift b in \(L_2\) L 2 in the case if the drift satisfies the “non-spectral” condition \(\operatorname {div} b\le 0\) div b ≤ 0 .