<p>In this paper, we consider the following anisotropic quasi-geostrophic equations <Equation ID="Equ28"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1205_Article_Equ28.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="515" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \partial _t\theta + u_\theta \cdot \nabla \theta +\mu |\partial _1|^{2\alpha }\theta +\nu |\partial _2|^{2\beta }\theta =0,\quad u_\theta =\mathcal {R}^{\perp }\theta , \qquad \qquad (AQG)_{\alpha ,\beta } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>θ</mi> <mo>+</mo> <msub> <mi>u</mi> <mi>θ</mi> </msub> <mrow> <mo>·</mo> <mi mathvariant="normal">∇</mi> <mi>θ</mi> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">|</mo> </mrow> <msub> <mi>∂</mi> <mn>1</mn> </msub> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </msup> <mi>θ</mi> <mo>+</mo> <mi>ν</mi> <msup> <mrow> <mo stretchy="false">|</mo> <msub> <mi>∂</mi> <mn>2</mn> </msub> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>2</mn> <mi>β</mi> </mrow> </msup> <mi>θ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <msub> <mi>u</mi> <mi>θ</mi> </msub> <mo>=</mo> <msup> <mrow> <mi mathvariant="script">R</mi> </mrow> <mo>⊥</mo> </msup> <mi>θ</mi> <mo>,</mo> <mspace width="2em" /> <mspace width="2em" /> <msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>Q</mi> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1205_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="103" /> </InlineMediaObject> <EquationSource Format="TEX">$\min \{\alpha ,\beta \}=\frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">min</mo> <mrow> <mo stretchy="false">{</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">}</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> et <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1205_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="138" /> </InlineMediaObject> <EquationSource Format="TEX">$\max \{\alpha ,\beta \}\in \left( \frac{1}{2},1\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">max</mo> <mrow> <mo stretchy="false">{</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">}</mo> </mrow> <mo>∈</mo> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. This equation is a particular case of the equation introduced by Ye (2019) in Ye (Nonlinearity 33:72, 2019). In this paper, we prove that for any initial data <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1205_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta ^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>θ</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> in the Sobolev space <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1205_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$H^{s}(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1205_Article_IEq5.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$s >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the equation <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1205_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$(AQG)_{\alpha ,\beta }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>Q</mi> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> has a global solution <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1205_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1205_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="114" /> </InlineMediaObject> <EquationSource Format="TEX">$C_b(\mathbb {R}^+,H^s(\mathbb {R}^2)).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>b</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> <mo>,</mo> <msup> <mi>H</mi> <mi>s</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> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

Global regularity of semi-critical case of anisotropic quasi-geostrophic equations in Sobolev spaces

期刊论文
发表日期：

2025年3月10日

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Mustapha Amara

摘要

In this paper, we consider the following anisotropic quasi-geostrophic equations $\begin{aligned} \partial _t\theta + u_\theta \cdot \nabla \theta +\mu |\partial _1|^{2\alpha }\theta +\nu |\partial _2|^{2\beta }\theta =0,\quad u_\theta =\mathcal {R}^{\perp }\theta , \qquad \qquad (AQG)_{\alpha ,\beta } \end{aligned}$ $\begin{matrix} \partial_{t} θ + u_{θ} \cdot \nabla θ + μ | \partial_{1} |^{2 α} θ + ν {| \partial_{2} |}^{2 β} θ = 0, u_{θ} = R^{⊥} θ, {(A Q G)}_{α, β} \end{matrix}$ where $\min \{\alpha ,\beta \}=\frac{1}{2}$ $min {α, β} = \frac{1}{2}$ et $\max \{\alpha ,\beta \}\in \left( \frac{1}{2},1\right) $ $max {α, β} \in (\frac{1}{2}, 1)$ . This equation is a particular case of the equation introduced by Ye (2019) in Ye (Nonlinearity 33:72, 2019). In this paper, we prove that for any initial data $\theta ^0$ $θ^{0}$ in the Sobolev space $H^{s}(\mathbb {R}^2)$ $H^{s} (R^{2})$ , $s >1$ $s > 1$ , the equation $(AQG)_{\alpha ,\beta }$ ${(A Q G)}_{α, β}$ has a global solution $\theta $ $θ$ in $C_b(\mathbb {R}^+,H^s(\mathbb {R}^2)).$ $C_{b} (R^{+}, H^{s} (R^{2})) .$