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<p>We investigate the long-time behavior of large solutions to the compressible Navier–Stokes equations subject to large external potential forces in a three-dimensional (3D) bounded domain with slip boundary conditions. We demonstrate that large solutions converge to non-constant steady states (induced by the large external force) at an exponential rate 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 bounded above and the initial potential energy is properly small. Furthermore, under the additional assumption that the initial density is bounded below, we establish the exponential convergence of the density to its steady state 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. Additionally, we show that vacuum states persist, and the derivative of the density must blow up at an exponential rate in the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^r\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation>-norm (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(r &gt; 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>&gt;</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>) as time approaches infinity if the initial density includes a vacuum.</p>

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Long-Time Behavior to the Compressible Navier–Stokes Equations Subject to Large External Forces in 3D Bounded Domains

  • 期刊论文

  • 发表日期：

    2026年5月12日

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  • Lin Xu,
  • Xin Zhong

摘要

We investigate the long-time behavior of large solutions to the compressible Navier–Stokes equations subject to large external potential forces in a three-dimensional (3D) bounded domain with slip boundary conditions. We demonstrate that large solutions converge to non-constant steady states (induced by the large external force) at an exponential rate in the \(L^2\) L 2 -norm, provided the density is essentially bounded above and the initial potential energy is properly small. Furthermore, under the additional assumption that the initial density is bounded below, we establish the exponential convergence of the density to its steady state in the \(L^\infty \) L ∞ -norm. Additionally, we show that vacuum states persist, and the derivative of the density must blow up at an exponential rate in the \(L^r\) L r -norm ( \(r > 3\) r > 3 ) as time approaches infinity if the initial density includes a vacuum.