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<p>We investigate the global stability of large solutions to the compressible isentropic Navier–Stokes equations in a three-dimensional (3D) bounded domain with Navier-slip boundary conditions. It is shown that the strong solutions converge to an equilibrium state exponentially in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm provided the density is essentially uniform-in-time bounded from above. Moreover, we obtain that the density converges to its equilibrium state exponentially in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-norm if additionally the initial density is bounded away from zero. Furthermore, we derive that the vacuum states will not vanish for any time provided vacuum appears (even at a point) initially. This is the first result concerning the global stability for large strong solutions of compressible Navier–Stokes equations with vacuum in 3D general bounded domains.</p>

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Global Stability for Compressible Isentropic Navier–Stokes Equations in 3D Bounded Domains with Navier-slip Boundary Conditions

  • 期刊论文

  • 发表日期：

    2026年6月22日

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  • Yang Liu,
  • Guochun Wu,
  • Xin Zhong

摘要

We investigate the global stability of large solutions to the compressible isentropic Navier–Stokes equations in a three-dimensional (3D) bounded domain with Navier-slip boundary conditions. It is shown that the strong solutions converge to an equilibrium state exponentially in the \(L^2\) L 2 -norm provided the density is essentially uniform-in-time bounded from above. Moreover, we obtain that the density converges to its equilibrium state exponentially in the \(L^\infty \) L ∞ -norm if additionally the initial density is bounded away from zero. Furthermore, we derive that the vacuum states will not vanish for any time provided vacuum appears (even at a point) initially. This is the first result concerning the global stability for large strong solutions of compressible Navier–Stokes equations with vacuum in 3D general bounded domains.