<p>We investigate the zero-Mach limit of compressible Navier-Stokes equations in 3D bounded domains with non-slip boundary condition. No smallness restrictions are imposed on the initial velocity and the time interval. If the limiting system admits a reasonably smooth solution on a certain period, we verify that the corresponding compressible system admits the smooth solution on the same duration as well, provided the Mach number is small enough. Moreover, the solutions of compressible system converge uniformly to that of the incompressible one as Mach number tends to zero. We apply the global geometric tools introduced by Chrisrodoulou-Lindblad [8] to get higher order estimates of the density near the boundary, which also help us relax the smallness condition <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Vert \nabla ^2\rho _0\Vert \le C\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mn>2</mn> </msup> <msub> <mi>ρ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">‖</mo> <mo>≤</mo> <mi>C</mi> <mi>ε</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> in previous works to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Vert \nabla ^2\rho _0\Vert \le C\varepsilon ^{-\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mn>2</mn> </msup> <msub> <mi>ρ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">‖</mo> <mo>≤</mo> <mi>C</mi> </mrow> <msup> <mi>ε</mi> <mrow> <mo>-</mo> <mi>α</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

The Zero-Mach Limit of Compressible Navier-Stokes Equations in Bounded Domains with Non-slip Boundary Condition

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发表日期：

2025年11月17日

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Xinyu Fan,
Qiangchang Ju,
Jianjun Xu

摘要

We investigate the zero-Mach limit of compressible Navier-Stokes equations in 3D bounded domains with non-slip boundary condition. No smallness restrictions are imposed on the initial velocity and the time interval. If the limiting system admits a reasonably smooth solution on a certain period, we verify that the corresponding compressible system admits the smooth solution on the same duration as well, provided the Mach number is small enough. Moreover, the solutions of compressible system converge uniformly to that of the incompressible one as Mach number tends to zero. We apply the global geometric tools introduced by Chrisrodoulou-Lindblad [8] to get higher order estimates of the density near the boundary, which also help us relax the smallness condition $\Vert \nabla ^2\rho _0\Vert \le C\varepsilon $ $‖ \nabla^{2} ρ_{0} ‖ \leq C ε$ in previous works to $\Vert \nabla ^2\rho _0\Vert \le C\varepsilon ^{-\alpha }$ $‖ \nabla^{2} ρ_{0} ‖ \leq C ε^{- α}$ for some $\alpha \ge 0$ $α \geq 0$ .