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<p>We consider the global existence and large-time behaviour of strong solutions to the Dirichlet problem of three-dimensional inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity. When the viscosity coefficient <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mu (\rho )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a power function of the density (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mu (\rho )=\mu \rho ^\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>μ</mi> <msup> <mi>ρ</mi> <mi>α</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha &gt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>), after fully using the dissipative structure of the system, we obtain the global existence and exponential behaviour of the strong solution, provided that the initial density is sufficiently large. It should be noticed that the initial velocity can be arbitrarily large. This is the first result concerning the well-posedness of large strong solution for the inhomogeneous Navier-Stokes equations in three dimensions without smallness of the velocity.</p>

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Existence and exponential stability of the global large strong solution to the 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity and large velocity

  • 期刊论文

  • 发表日期：

    2026年1月12日

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  • Xiangdi Huang,
  • Jiaxu Li,
  • Rong Zhang

摘要

We consider the global existence and large-time behaviour of strong solutions to the Dirichlet problem of three-dimensional inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity. When the viscosity coefficient \(\mu (\rho )\) μ ( ρ ) is a power function of the density ( \(\mu (\rho )=\mu \rho ^\alpha \) μ ( ρ ) = μ ρ α with \(\alpha >1\) α > 1 ), after fully using the dissipative structure of the system, we obtain the global existence and exponential behaviour of the strong solution, provided that the initial density is sufficiently large. It should be noticed that the initial velocity can be arbitrarily large. This is the first result concerning the well-posedness of large strong solution for the inhomogeneous Navier-Stokes equations in three dimensions without smallness of the velocity.