<p>In this paper, we study the global well-posedness of isentropic compressible Navier-Stokes equations in three-dimensional (3<i>D</i>) periodic thin domains of type <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb{T}\times(\delta\mathbb{T})^{2}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mi>δ</mi> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, where 0 &lt; <i>δ</i> &lt; 1 is a small parameter. We apply Littlewood-Paley decomposition theory to the periodic thin domain <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb{T}\times(\delta\mathbb{T})^{2}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mi>δ</mi> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> and show some Bernstein type inequalities with specific dependence on the parameter <i>δ</i>. This allows us to establish various embedding inequalities in Besov spaces in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb{T}\times(\delta\mathbb{T})^{2}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mi>δ</mi> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> as well as the interpolation inequalities of Gagliardo-Nirenberg type. Together with ideas of Hoff (1995), we prove that the compressible Navier-Stokes equations in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb{T}\times(\delta\mathbb{T})^{2}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mi>δ</mi> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> admit a unique global regular solution when the thickness <i>δ</i> of the domain is sufficiently small, even if the initial data (<i>ρ</i><sub>0,<i>δ</i></sub>, <Emphasis Type="BoldItalic">u</Emphasis><sub>0,<i>δ</i></sub>) are large in the sense that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\Vert(\nabla^{2}\rho_{0},\nabla^{2}u_{0,\delta})\Vert_{L^{2}(\mathbb{T}\times(\delta\mathbb{T})^{2})}\sim\delta^{-\kappa}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo fence="false" stretchy="false">∥</mo> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow> <mn>2</mn> </mrow> </msup> <msub> <mi>ρ</mi> <mrow> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow> <mn>2</mn> </mrow> </msup> <msub> <mi>u</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>δ</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo>∥</mo> <mrow> <msup> <mi>L</mi> <mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mi>δ</mi> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msub> <mo>∼</mo> <msup> <mi>δ</mi> <mrow> <mo>−</mo> <mi>κ</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\kappa\in(0,{1\over{2}})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>κ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>.</p>

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Global solutions to isentropic compressible Navier-Stokes equations in 3D thin domains

  • Sai Li,
  • Yong Lu,
  • Yongzhong Sun

摘要

In this paper, we study the global well-posedness of isentropic compressible Navier-Stokes equations in three-dimensional (3D) periodic thin domains of type \(\mathbb{T}\times(\delta\mathbb{T})^{2}\) T × ( δ T ) 2 , where 0 < δ < 1 is a small parameter. We apply Littlewood-Paley decomposition theory to the periodic thin domain \(\mathbb{T}\times(\delta\mathbb{T})^{2}\) T × ( δ T ) 2 and show some Bernstein type inequalities with specific dependence on the parameter δ. This allows us to establish various embedding inequalities in Besov spaces in \(\mathbb{T}\times(\delta\mathbb{T})^{2}\) T × ( δ T ) 2 as well as the interpolation inequalities of Gagliardo-Nirenberg type. Together with ideas of Hoff (1995), we prove that the compressible Navier-Stokes equations in \(\mathbb{T}\times(\delta\mathbb{T})^{2}\) T × ( δ T ) 2 admit a unique global regular solution when the thickness δ of the domain is sufficiently small, even if the initial data (ρ0,δ, u0,δ) are large in the sense that \(\Vert(\nabla^{2}\rho_{0},\nabla^{2}u_{0,\delta})\Vert_{L^{2}(\mathbb{T}\times(\delta\mathbb{T})^{2})}\sim\delta^{-\kappa}\) ( 2 ρ 0 , 2 u 0 , δ ) L 2 ( T × ( δ T ) 2 ) δ κ with \(\kappa\in(0,{1\over{2}})\) κ ( 0 , 1 2 ) .