<p>We study the analyticity and Gevrey regularity for solutions to the 2-dimensional subcritical dissipative generalized surface quasi-geostrophic (gSQG) equations. Motivated by the works of Chemin et al. (Math Res Lett 27:1631–1643, 2020) and Herbst and Skibsted (Adv Math 228(4):1990–2033, 2011) concerning the analyticity radius of solutions to the classical Navier–Stokes system, we prove in this paper that the radius of analyticity of the solutions to the subcritical dissipative generalized SQG system in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3692_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> can be improved to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3692_Article_IEq2.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$Ct^{\frac{1}{\kappa }}(-\ln t)^{1-\frac{1}{\kappa }}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <msup> <mi>t</mi> <mfrac> <mn>1</mn> <mi>κ</mi> </mfrac> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mo>ln</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mfrac> <mn>1</mn> <mi>κ</mi> </mfrac> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for small time. Moreover, in the periodic case, we can improve the radius of analyticity to <i>Ct</i>.</p>

Gevrey regularity for the subcritical dissipative generalized surface quasi-geostrophic equations

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发表日期：

2025年2月25日

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Wen Deng,
Marius Paicu

摘要

We study the analyticity and Gevrey regularity for solutions to the 2-dimensional subcritical dissipative generalized surface quasi-geostrophic (gSQG) equations. Motivated by the works of Chemin et al. (Math Res Lett 27:1631–1643, 2020) and Herbst and Skibsted (Adv Math 228(4):1990–2033, 2011) concerning the analyticity radius of solutions to the classical Navier–Stokes system, we prove in this paper that the radius of analyticity of the solutions to the subcritical dissipative generalized SQG system in $\mathbb {R}^2$ $R^{2}$ can be improved to $Ct^{\frac{1}{\kappa }}(-\ln t)^{1-\frac{1}{\kappa }}$ $C t^{\frac{1}{κ}} {(- ln t)}^{1 - \frac{1}{κ}}$ for small time. Moreover, in the periodic case, we can improve the radius of analyticity to Ct.