arXiv subjects

Muxi Lei

Publications and source records attributed to Muxi Lei.

3 recordsLinked to original sources

arXiv2026

On the Cauchy problem for the multi-dimensional compressible Navier-Stokes-Korteweg system: Global strong solutions with arbitrarily large initial data

Since the pioneering work of Korteweg (1901) and the subsequent refinement of capillary fluid models by Dunn and Serrin (1985), the global existence of strong solutions to the multi-dimensional compressible Navier-Stokes-Korteweg (NSK) system with arbitrarily large initial data has stood as a formidable open problem in fluid mechanics. This challenge was recently addressed by [Gu-Huang-Meng-Zhou, arXiv:2603.11762], who established the global existence of strong solutions for arbitrarily large initial data on the periodic domain $\mathbb{T}^N$ ($N=2,3$), provided that the viscosity coefficients satisfy a BD-type algebraic relation ($μ(ρ) = νρ^α, λ(ρ) = 2ν(α-1)ρ^α$) and the Korteweg stress tensor complies with a generalized Bohm identity ($κ(ρ) = \varepsilon^2 α^2 ρ^{2α-3}$). However, the existence of global strong solutions for the Cauchy problem under these conditions has remained an open question. In this paper, we resolve this problem by proving the global existence of strong solutions for the Cauchy problem ($\mathbb{R}^N$, $N=2,3$) with arbitrarily large initial data and non-vacuum far-field density. By employing a refined truncation analysis combined with an original modified Nash-Moser type iteration scheme, we overcome the difficulties arising from the lack of integrability for the density in the whole space. This result extends the large-data theory of compressible Navier-Stokes-Korteweg equations from bounded torus $\mathbb{T}^N$ to unbounded whole space $\mathbb{R}^N$, thus applicable to more general physical settings.

math.AP↗

arXiv2026

Global existence of classical solutions for the multi-dimensional compressible Navier-Stokes-Poisson equations on solid balls for arbitrary spherically symmetric large initial data

Whether the 3D compressible Navier-Stokes-Poisson equations admit global classical solutions for general large initial data has long been a challenging open problem. In this paper, we provide an affirmative answer to this question under spherical symmetry on solid balls . Specifically, we consider the initial-boundary value problem for the multi-dimensional compressible equations with density-dependent viscosity coefficients satisfying the BD-type entropy equality, namely, assuming $μ=ρ^α,\ λ=(α-1)ρ^α$ with $N=2, α\in (\frac{1}{2},1]$ and $N=3, α\in (\frac{5}{6},1]$, we establish the global existence of spherically symmetric classical solutions to the compressible Navier-Stokes-Poisson equations for both gaseous stars and plasmas with arbitrarily large initial data on solid balls. Our key observation lies in successfully handling the singularity at the center of the ball. By controlling the growth orders of the density and the gravitational potential at the central singularity, leveraging the structural advantages of the BD entropy and spherical symmetry, and fully exploiting the coupling between the effective velocity and the velocity, we establish $L^\infty$ estimates for the key quantities, which in turn yield upper and lower bound estimates for the density. This can be regarded as the first result on the existence of global classical solutions for arbitrarily large initial data to the compressible Navier-Stokes-Poisson equations in a truly multi-dimensional domain with high-dimensional features.

math.AP↗

arXiv2026

Global strong solutions with large initial data for the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Korteweg system

In this paper, we establish global strong solutions for arbitrarily large initial data to the 2D and 3D compressible Navier-Stokes-Korteweg system, also referred to as the quantum Navier-Stokes equations, originally derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985]. Specifically, we prove the existence of global strong solutions for arbitrarily large initial data in the case $N=2$ when $γ\ge 1$, and $N=3$ with $1 \le γ< 8/3$ for the associated Cauchy problem. By employing techniques from Littlewood-Paley theory, range truncation analysis, refined Nash-Moser and De Giorgi iteration methods, we derive positive upper and lower bounds for the density. As a consequence, we are able to treat the whole-space case with strictly positive far-field density. To the best of our knowledge, this is the first result that establishes global strong solutions for physically relevant compressible Navier-Stokes equations in the whole space, without imposing any symmetry or special geometric assumptions on the initial data.

math.AP↗