<p>This paper investigates the global stability of large solutions to the compressible Navier–Stokes–Poisson system in a three-dimensional bounded domain with Navier-type slip boundary conditions, allowing large non-flat doping profiles and initial density with vacuum states. We prove exponential convergence of solutions to the steady state 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 has an essential upper bound. Additionally, if the initial density is bounded below by a positive constant, the density converges exponentially to its non-constant steady state in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-norm. Conversely, when the initial density contains a vacuum (even at a single point), we show that vacuum states persist, and the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$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 > 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>) of the density derivatives blow up as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$t \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. A key novelty of our results is the admission of steady states and doping profiles with large spatial variations, without smallness of initial energy. To our knowledge, this is the first result establishing global stability of large solutions to the compressible Navier–Stokes–Poisson system with vacuum in general three-dimensional bounded domains.</p>

Global Stability of Large Solutions to the Compressible Navier–Stokes–Poisson System with Large Non-Flat Doping Profiles in Three-Dimensional Bounded Domains

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发表日期：

2026年3月7日

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Lin Xu

摘要

This paper investigates the global stability of large solutions to the compressible Navier–Stokes–Poisson system in a three-dimensional bounded domain with Navier-type slip boundary conditions, allowing large non-flat doping profiles and initial density with vacuum states. We prove exponential convergence of solutions to the steady state in the $L^2$ $L^{2}$ -norm, provided the density has an essential upper bound. Additionally, if the initial density is bounded below by a positive constant, the density converges exponentially to its non-constant steady state in the $L^\infty $ $L^{\infty}$ -norm. Conversely, when the initial density contains a vacuum (even at a single point), we show that vacuum states persist, and the $L^r$ $L^{r}$ -norm ( $ r > 3$ $r > 3$ ) of the density derivatives blow up as $t \rightarrow \infty $ $t \to \infty$ . A key novelty of our results is the admission of steady states and doping profiles with large spatial variations, without smallness of initial energy. To our knowledge, this is the first result establishing global stability of large solutions to the compressible Navier–Stokes–Poisson system with vacuum in general three-dimensional bounded domains.