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<p>In this article, we are concerned with the one-dimensional full compressible Navier–Stokes–Korteweg equations, which govern the motions of compressible viscous fluids with internal capillarity and heat-conductive. We focus on the low Mach number limit for the one-dimensional full compressible Navier–Stokes–Korteweg equations with well-prepared and ill-prepared data, whose density and temperature have different asymptotic states at infinity. That the solutions of the one-dimensional full compressible Navier–Stokes–Korteweg equations with the well-prepared data converge to a nonlinear diffusion wave solution globally in time as Mach number goes to zero with the convergence rate, is shown when the difference between the states at <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\pm \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>±</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> is suitably small. In particular, we note that the velocity of diffusion wave is only driven by the variation of temperature, and the solution of the one-dimensional full compressible Navier–Stokes–Koteweg system also has the same property when Mach number is small. Next, the Mach limit for the ill-prepared data is proven by the uniform estimates including weighted time derivatives and an extended convergence lemma, when the difference between the states at <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\pm \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>±</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> can be arbitrary large.</p>

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Low mach number limit of the one-dimensional full compressible Navier–Stokes–Korteweg equations

  • 期刊论文

  • 发表日期：

    2026年2月13日

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  • Yeping Li,
  • Rong Yin

摘要

In this article, we are concerned with the one-dimensional full compressible Navier–Stokes–Korteweg equations, which govern the motions of compressible viscous fluids with internal capillarity and heat-conductive. We focus on the low Mach number limit for the one-dimensional full compressible Navier–Stokes–Korteweg equations with well-prepared and ill-prepared data, whose density and temperature have different asymptotic states at infinity. That the solutions of the one-dimensional full compressible Navier–Stokes–Korteweg equations with the well-prepared data converge to a nonlinear diffusion wave solution globally in time as Mach number goes to zero with the convergence rate, is shown when the difference between the states at \(\pm \infty \) ± ∞ is suitably small. In particular, we note that the velocity of diffusion wave is only driven by the variation of temperature, and the solution of the one-dimensional full compressible Navier–Stokes–Koteweg system also has the same property when Mach number is small. Next, the Mach limit for the ill-prepared data is proven by the uniform estimates including weighted time derivatives and an extended convergence lemma, when the difference between the states at \(\pm \infty \) ± ∞ can be arbitrary large.