[論文レビュー] Global weak solutions to the isentropic compressible Navier-Stokes equations with vacuum and unbounded density in a half-plane under Dirichlet boundary conditions

Shuai Wang, Xin Zhong|arXiv (Cornell University)|Jan 17, 2026

Navier-Stokes equation solutions被引用数 0

ひとこと要約

tldr: The paper proves the global existence of a class of weak solutions to the isentropic compressible Navier–Stokes equations in a half-plane with Dirichlet boundary conditions, allowing vacuum and unbounded density under a small initial energy regime. The solutions have intermediate regularity between Lions–Feireist and Hoff frameworks and rely on Green function techniques for boundary handling.

ABSTRACT

We establish the global existence of a class of weak solutions to the isentropic compressible Navier-Stokes equations in a half-plane with Dirichlet boundary conditions, allowing for vacuum both in the interior and at infinity, under a suitably small initial total energy. The solutions constructed here admit unbounded densities and lie in an intermediate regularity regime between the finite-energy weak solutions of Lions-Feireisl and the framework of Hoff. This result generalizes previous works of Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365-1407) and Perepelitsa (Arch. Ration. Mech. Anal. 212 (2014), pp. 709-726) concerning discontinuous solutions by allowing vacuum states and unbounded density. Our analysis relies on the Green function method and new estimates involving the specific structure of the equations and the geometry of the half-plane. To the best of our knowledge, this is the first result concerning global weak solutions within Hoff's framework on an unbounded domain that simultaneously accommodates Dirichlet boundary conditions and far-field vacuum. The intermediate-regularity class developed here may be viewed as a natural extension of Hoff's theory, precisely tailored to overcome the two corresponding obstructions: the lack of global space-time control of the effective viscous flux arising from far-field vacuum and the absence of boundary-induced regularity gains in the no-slip setting.

研究の動機と目的

• Motivate the study of global weak solutions to isentropic compressible Navier–Stokes with vacuum in unbounded domains and Dirichlet boundaries.
• Establish a global existence result for a class of weak solutions with density possibly unbounded and vacuum at interior and infinity.
• Develop an intermediate regularity framework between Lions–Feireisl and Hoff theories suitable for half-plane domains.
• Overcome lack of boundary-induced regularity and far-field decay to control nonlinear terms.
• Provide a strategy that blends Green function decomposition with density integrability to achieve a priori bounds.

提案手法

• Formulate weak solutions in the half-plane with Dirichlet boundary conditions.
• Introduce the effective viscous flux F and decompose it using Green function techniques.
• Establish a priori estimates in an intermediate L^p framework with P(ρ) ∈ L^q and ρ ∈ L^θ.
• Split the velocity into pieces and employ L^6 estimates to handle nonlinear terms.
• Utilize Zlotnik’s inequality to obtain time-uniform upper bounds for ρ in L^θ.
• Construct global solutions via a two-step limiting procedure from strictly positive density to vanishing lower bound.

実験結果

リサーチクエスチョン

• RQ1 Can global weak solutions be constructed for the isentropic compressible Navier–Stokes equations in a half-plane with Dirichlet boundary conditions and far-field vacuum?
• RQ2 Is it possible to allow vacuum both in the interior and at infinity while permitting unbounded densities within an intermediate regularity class?
• RQ3 How can Green function methods and an L^p density framework be combined to control the effective viscous flux and nonlinear terms under no-slip boundaries?
• RQ4 What regularity and integrability on the pressure and density suffice to close the a priori estimates and enable global existence?

主な発見

• A global weak solution exists under a small initial total energy for the half-plane problem with Dirichlet boundary conditions.
• The constructed solutions admit unbounded densities and lie in an intermediate regularity regime between Lions–Feireisl and Hoff.
• A decomposition of the effective viscous flux via Green functions yields key estimates under Dirichlet boundary conditions.
• The pressure P(ρ) is shown to belong to L^q for suitable q, enabling improved control of the far-field behavior.
• A two-step limiting procedure from positive-density approximations to vacuum is used to obtain global existence.