arXiv subjects

Weili Meng

Publications and source records attributed to Weili Meng.

7 recordsLinked to original sources

arXiv2026

On the large-time behavior of strong solutions to the generalized compressible Navier-Stokes-Korteweg system in 2D and 3D for arbitrarily large initial data

In this paper, we establish the global existence and large-time behavior of strong solutions for the two- and three-dimensional periodic compressible Navier-Stokes-Korteweg system with arbitrarily large initial data $(\rho_0,u_0)\in H^3\times H^2$. The viscosity coefficients satisfy the BD relation $\mu(\rho)=\nu\rho^\alpha$ and $\lambda(\rho)=2\nu(\alpha-1)\rho^\alpha$, while the capillarity coefficient is given by $\kappa(\rho)=\varepsilon^2\alpha^2\rho^{2\alpha-3}$. For the case $\alpha<1$, we first enlarge the admissible parameter range for the global existence of strong solutions established in Gu-Huang-Meng-Zhou [arXiv:2603.11762 (2026)] by exploiting the doubly parabolic structure of the density-effective velocity system. We then develop a time-discretization strategy to establish uniform integrability estimates for the effective velocity, yielding a uniform upper bound for the density. Furthermore, we introduce a novel bootstrap argument to successively improve these integrability estimates, which leads to a uniform positive lower bound for the density. Finally, we derive global-in-time higher-order estimates and prove the large-time behavior \[ \left\|\rho(t)-\frac{1}{|\mathbb{T}^N|}\int_{\mathbb T^N}\rho_0 dx\right\|_{H^3} +\|\nabla u(t)\|_{H^1} \longrightarrow0, \qquad t\to\infty, \] without imposing any smallness assumption on the initial data. For the critical case $\alpha=1$, we improve the admissible parameter range established in Huang-Meng-Zhang [arXiv:2602.00455 (2026)] and establish a uniform upper bound for the density.

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arXiv2026

Global-in-time strong solutions for the 2D and 3D generalized compressible Navier-Stokes-Korteweg system with arbitrarily large initial data

In 1901, Korteweg formulated a constitutive equation for the Cauchy stress tensor to provide a continuum mechanical model for capillarity within fluids. Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133,1985] in 1985 further modified the system of compressible fluids based on the Korteweg theory of capillarity. Since then, for the 2D and 3D compressible Navier-Stokes-Korteweg system, the global existence of strong solutions with arbitrarily large initial data have remained a challenging open problem. In this paper, we provide an affirmative answer to this longstanding open problem. Specifically, under the assumption that the viscosity coefficients satisfy a BD-type algebraic relation of the form $\mu(\rho)=\nu\rho^{\alpha}$ and $\lambda(\rho)=2\nu(\alpha-1)\rho^{\alpha}$, and that the Korteweg stress tensor complies with a generalized Bohm identity of the form $\kappa(\rho)=\varepsilon^2\alpha^2\rho^{2\alpha-3}$, we establish the global existence of strong solutions for the 2D and 3D systems in torus with arbitrarily large regular initial data. The analysis is carried out in the intermediary non-dispersive regime, characterized by the condition that the capillarity coefficient constant $\varepsilon$ does not exceed the viscosity constant $\nu$. This result provides the first proof of the global-in-time existence of strong solutions for the 3D general Navier-Stokes-Korteweg system with arbitrarily large initial data in the non-dispersive regime.

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arXiv2026

Global regularity of the multi-dimensional compressible Navier-Stokes-Korteweg system with large initial data

In this work, we establish the global existence of strong solutions to the 2D and 3D compressible Navier-Stokes-Korteweg system with arbitrarily large initial data on the torus. This system was derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985] and is widely used to model capillarity in compressible fluids. Via an original modified Nash-Moser type iteration, we establish a critical novel estimate linking the effective velocity and the lower bound of the density, which plays a crucial role in deriving the positive lower bound of the density. To our knowledge, this can be viewed as the first existence result of global strong solutions for the compressible fluid dynamics equations with physical significance in general three-dimensional domains with arbitrarily large initial data.

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arXiv2025

On global classical and weak solutions with arbitrary large initial data to the multi-dimensional viscous Saint-Venant system and compressible Navier-Stokes equations subject to the BD entropy condition under spherical symmetry

In 1871, Saint-Venant introduced the renowned shallow water equations. Since then, for the two-dimensional viscous or inviscid shallow water equations, the global existence of smooth solutions with arbitrarily large initial data has remained a challenging and long-standing open problem. In this paper, we provide an affirmative resolution to the viscous problem under the assumption of two-dimensional radial symmetry. Specifically, we establish the global existence of smooth solutions for the two-dimensional radially symmetric viscous shallow water equations with arbitrary smooth initial data. To achieve this goal, our approach relies crucially on overcoming two major obstacles: first, treating the viscous Saint-Venant system as the endpoint case of the BD entropy condition for the compressible Navier-Stokes equations; and second, addressing the critical embedding imposed by the spatial dimension, which currently holds only in two dimensions. However, the same result can be extended to three dimension for the compressible Navier-Stokes equations satisfying general BD entropy conditions excluding the endpoint case. Indeed, under the same symmtric framework, we also prove the global existence of smooth solutions for arbitrarily large initial data for both the two- and three-dimensional compressible Navier-Stokes equations subject to the BD entropy condition. It is particularly noteworthy that the aforementioned shallow water equations precisely correspond to the endpoint case of the compressible Navier-Stokes equations satisfying the BD entropy condition.

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arXiv2025

Global well-posedness for 2D compressible radially symmetric Navier-Stokes equations with swirl

In this paper, we consider the radially symmetric compressible Navier-Stokes equations with swirl in two-dimensional disks, where the shear viscosity coefficient $\mu = \text{const}> 0$, and the bulk one $\lambda = \rho^\beta(\beta>0)$. When $\beta \geq 1$, we prove the global existence and asymptotic behavior of the large strong solutions for initial values that allow for vacuum. One of the key ingredients is to show the uniform boundedness of the density independent of the time. When $\beta\in(0,1)$, we prove the same conclusion holds when the initial value satisfies $\norm{\rho_0}_{L^\infty} \leq a_0$, where $a_0$ is given by \eqref{def a_0} as in Theorem \ref{Thm3}. To the best of our knowledge, this is the first result on the global existence of large strong solutions for 2D compressible Navier-Stokes equation with real non-slip (non Navier-slip) boundary conditions when $\beta\ge1$ and the first result on the global existence of strong solutions when $\beta\in(0,1)$

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arXiv2024

Free boundary value problem for the radial symmetric compressible isentropic Navier-Stokes equations with density-dependent viscosity

This paper is devoted to the study of free-boundary-value problem of the compressible Naiver-Stokes system with density-dependent viscosities $\mu=const>0,\lambda=\rho^\beta$ which was first introduced by Vaigant-Kazhikhov \cite{1995 Vaigant-Kazhikhov-SMJ} in 1995. By assuming the endpoint case $\beta=1$ in the radially spherical symmetric setting, we prove the (a priori) expanding rate of the free boundary is algebraic for multi-dimensional flow, and particularly establish the global existence of strong solution of the two-dimensional system for any large initial data. This also improves the previous work of Li-Zhang \cite{2016 Li-Zhang-JDE} where they proved the similar result for $\beta>1$. The main ingredients of this article is making full use of the geometric advantange of domain as well as the critical space dimension two.

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arXiv2022

Asymptotic mean value properties for the elliptic and parabolic double phase equations

We characterize an asymptotic mean value formula in the viscosity sense for the double phase elliptic equation $$ -{\rm div}(\lvert \nabla u \rvert^{p-2}\nabla u+ a(x)\lvert\nabla u \rvert^{q-2}\nabla u)=0 $$ and the normalized double phase parabolic equation $$ u_t=\lvert\nabla u \rvert ^{2-p}{\rm div}(\lvert \nabla u \rvert^{p-2}\nabla u+ a(x,t)\lvert\nabla u \rvert^{q-2}\nabla u), \quad 1<p\leq q<\infty. $$ This is the first mean value result for such kind of nonuniformly elliptic and parabolic equations. In addition, the results obtained can also be applied to the $p(x)$-Laplace equations and the variable coefficient $p$-Laplace type equations.

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