<p>In the recent work [arXiv:2308.03216], Coghi and Maurelli proved pathwise uniqueness of solutions to the vorticity form of stochastic 2D Euler equation, with Kraichnan transport noise and initial data in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_943_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$L^1\cap L^p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mo>∩</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_943_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$p>3/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. The aim of this note is to remove the constraint on <i>p</i>, showing that pathwise uniqueness holds for all <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_943_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$L^1\cap L^p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mo>∩</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> initial data with arbitrary <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_943_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the Pathwise Uniqueness of Stochastic 2D Euler Equations with Kraichnan Noise and $L^p$-data

期刊论文
发表日期：

2025年5月14日

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Shuaijie Jiao,
Dejun Luo

摘要

In the recent work [arXiv:2308.03216], Coghi and Maurelli proved pathwise uniqueness of solutions to the vorticity form of stochastic 2D Euler equation, with Kraichnan transport noise and initial data in $L^1\cap L^p$ $L^{1} \cap L^{p}$ for $p>3/2$ $p > 3 / 2$ . The aim of this note is to remove the constraint on p, showing that pathwise uniqueness holds for all $L^1\cap L^p$ $L^{1} \cap L^{p}$ initial data with arbitrary $p>1$ $p > 1$ .

On the Pathwise Uniqueness of Stochastic 2D Euler Equations with Kraichnan Noise and \(L^p\)-data

摘要