<p>In this paper, we establish the existence of global self-similar solutions to the 3D Muskat equation when the two fluids have the same viscosity but different densities. These self-similar solutions are globally defined in both space and time, with exact cones as their initial data. Furthermore, we estimate the difference between our self-similar solutions and solutions of the linearized equation around the flat interface in terms of critical spaces and some weighted <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2126_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$\dot{W}^{k,\infty }(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mover accent="true"> <mi>W</mi> <mo>˙</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>∞</mi> </mrow> </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> spaces for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2126_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$k=1,2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. The main ingredients of the proof are new estimates in the sense of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2126_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="132" /> </InlineMediaObject> <EquationSource Format="TEX">$\dot{H}^{s_1}(\mathbb {R}^2) \cap \dot{H}^{s_2}(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <msub> <mi>s</mi> <mn>1</mn> </msub> </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>∩</mo> <msup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <msub> <mi>s</mi> <mn>2</mn> </msub> </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> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2126_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="161" /> </InlineMediaObject> <EquationSource Format="TEX">$3/2<s_1<2<s_2<3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo><</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo><</mo> <mn>2</mn> <mo><</mo> <msub> <mi>s</mi> <mn>2</mn> </msub> <mo><</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, which is continuously embedded in critical spaces for the 3D Muskat problem: <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2126_Article_IEq5.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$\dot{H}^2(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mn>2</mn> </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> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2126_Article_IEq6.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$\dot{W}^{1,\infty }(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mover accent="true"> <mi>W</mi> <mo>˙</mo> </mover> <mrow> <mn>1</mn> <mo>,</mo> <mi>∞</mi> </mrow> </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>.</p>

Global Self-Similar Solutions for the 3D Muskat Equation

期刊论文
发表日期：

2025年8月19日

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Jungkyoung Na

摘要

In this paper, we establish the existence of global self-similar solutions to the 3D Muskat equation when the two fluids have the same viscosity but different densities. These self-similar solutions are globally defined in both space and time, with exact cones as their initial data. Furthermore, we estimate the difference between our self-similar solutions and solutions of the linearized equation around the flat interface in terms of critical spaces and some weighted $\dot{W}^{k,\infty }(\mathbb {R}^2)$ ${\dot{W}}^{k, \infty} (R^{2})$ spaces for $k=1,2$ $k = 1, 2$ . The main ingredients of the proof are new estimates in the sense of $\dot{H}^{s_1}(\mathbb {R}^2) \cap \dot{H}^{s_2}(\mathbb {R}^2)$ ${\dot{H}}^{s_{1}} (R^{2}) \cap {\dot{H}}^{s_{2}} (R^{2})$ with $3/2<s_1<2<s_2<3$ $3 / 2 < s_{1} < 2 < s_{2} < 3$ , which is continuously embedded in critical spaces for the 3D Muskat problem: $\dot{H}^2(\mathbb {R}^2)$ ${\dot{H}}^{2} (R^{2})$ and $\dot{W}^{1,\infty }(\mathbb {R}^2)$ ${\dot{W}}^{1, \infty} (R^{2})$ .