<p>We establish the short-time existence and uniqueness of non-decaying solutions to the generalized Surface Quasi-Geostrophic equations in Hölder-Zygmund spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C^r(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>C</mi> <mi>r</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$r>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and uniformly local Sobolev spaces <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H_{ul}^s(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mrow> <mi mathvariant="italic">ul</mi> </mrow> <mi>s</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$s>2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Existence of Non-Decaying Solutions to the Generalized Surface Quasi-Geostrophic Equations

期刊论文
发表日期：

2025年11月12日

查看全文

Zachary Radke

摘要

We establish the short-time existence and uniqueness of non-decaying solutions to the generalized Surface Quasi-Geostrophic equations in Hölder-Zygmund spaces $C^r(\mathbb {R}^2)$ $C^{r} (R^{2})$ for $r>1$ $r > 1$ and uniformly local Sobolev spaces $H_{ul}^s(\mathbb {R}^2)$ $H_{ul}^{s} (R^{2})$ for $s>2$ $s > 2$ .