Searcharxiv⌕ Search

arXiv subjects

Jingchi Huang

Publications and source records attributed to Jingchi Huang.

9 recordsLinked to original sources
arXiv2026

Local Well-Posedness for Compressible Capillary-Gravity Water Waves with Acute Contact Angles

Our purpose is to investigate the local well-posedness of the compressible Euler equations in a two-dimensional bounded corner domain with acute contact angles. This configuration describes a free surface intersecting the fixed bottom at two points, where the fluid is subject to a gravitational field and the interface between the fluid and air is influenced by capillary forces. When the contact angles are less than $\pi/2$, we establish a local existence theory for the solution, with dissipation effects occurring at the contact points. The main analytical challenge arises from contact point singularities, which renders previous methods for dealing with compressible free boundary problems inadequate. To overcome this, we first establish the geometric structure for the compressible Euler equations, an approach originally introduced by Shatah and Zeng \cite{Shatah2008} for incompressible fluids. Additionally, we provide a singularity analysis for the wave equations in the corner domain, which ensures the validity of calculations near the corner. Finally, based on the geometric structure and singularity analysis, we obtain a priori energy estimates. Using these estimates, we also prove the local well-posedness of the system in a geometric formulation. To our knowledge, this is the first result addressing compressible Euler equations with a free boundary that involves contact points.

math.AP↗
arXiv2026

Long-time asymptotics and invariant manifold for the fractional 2D Navier-Stokes equation

We consider the two-dimensional incompressible Navier-Stokes equations with supercritical fractional dissipation in the vorticity formulation. In self-similar variables, we analyze the linearized operator in weighted spaces, prove a spectral gap, and construct a finite-dimensional local slow invariant manifold for small solutions. As a consequence, solutions are attracted to this manifold and admit an explicit long-time asymptotic expansion determined by the leading eigenmodes; additional moment conditions yield faster decay. We also show Lipschitz dependence of the manifold on the dissipation exponent, and recover the classical Navier-Stokes dynamics in the limit as the exponent approaches one.

math.AP↗
arXiv2024

The interaction between rough vortex patch and boundary layer

In this paper, we investigate the asymptotic behavior of solutions to the Navier-Stokes equations in the half-plane under high Reynolds number conditions, where the initial vorticity belongs to the Yudovich class and is supported away from the boundary. We establish the $L^p$ ($2\leq p< \infty$) convergence of solutions from the Navier-Stokes equations to those of the Euler equations. One of the main difficulties stems from the limited regularity of the initial data, which hinders the derivation of an asymptotic expansion. To overcome this challenge, we first prove a Kato-type criterion adapted to the Yudovich class setting. We then obtain uniform estimates for the Navier-Stokes equations -- a non-trivial task due to the strong boundary layer effects. A key component of our approach is the introduction of a suitable functional framework, which enables us to control the interaction between the rough vortex patch and the boundary layer.

math.AP↗
arXiv2020

Stability of large solutions for full compressible Navier-Stokes equations in the whole spaces

The current paper is devoted to the investigation of the global-in-time stability of large solutions for the full Navier-Stokes-Fourier system in the whole space. Suppose that the density and the temperature are bounded from above uniformly in time in the Holder space $C^α$ with $α$ sufficiently small and in $L^\infty$ space respectively. Then we prove two results: (1). Such kind of the solution will converge to its associated equilibrium with a rate which is the same as that for the heat equation if we impose the same condition on the initial data. As a result, we obtain the propagation of positive lower bounds of the density and the temperature. (2). Such kind of the solution is stable, that is, any perturbed solution will remain close to the reference solution if initially they are close to each other. This shows that the set of the smooth and bounded solutions is open.

math.AP↗
arXiv2017

Global stability of large solutions to the 3D compressible Navier-Stokes equations

The present paper aims at the investigation of the global stability of large solutions to the compressible Navier-Stokes equations in the whole space. Our main results and innovations can be concluded as follows: Under the assumption that the density $ρ(t,x)$ verifies $ρ(0,x)\ge c>0$ and $\sup_{t\ge0}\|ρ(t)\|_{C^α}\le M$ with $α$ sufficiently small, we establish a new mechanism for the convergence of the solution to its associated equilibrium with an explicit decay rate which is as the same as that for the heat equation. The main idea of the proof relies on the basic energy identity, techniques from blow-up criterion and a new estimate for the low frequency part of the solution. We prove the global-in-time stability for the equations, i.e, any perturbed solution will remain close to the reference solution if initially they are close to each other. Our result implies that the set of the smooth and bounded solutions is an open set. Going beyond the close-to-equilibrium setting, we construct the global large solutions to the equations with a class of initial data in $L^p$ type critical spaces. Here the "large solution" means that the vertical component of the velocity could be arbitrarily large initially.

math.AP↗
arXiv2012

Global solutions to 2-d inhomogeneous navier-stokes system with general velocity

In this paper, we are concerned with the global wellposedness of 2-D density-dependent incompressible Navier-Stokes equations with variable viscosity, in a critical functional frame- work which is invariant by the scaling of the equations and under a non-linear smallness condition on fluctuation of the initial density which has to be doubly exponential small compared with the size of the initial velocity. In the second part of the paper, we apply our methods combined with the techniques of R. Danchin and P. B. Mucha to prove the global existence of solutions to inhomogeneous Navier-Stokes system with piecewise constant initial density which has small jump at the interface and is away from vacuum. In particular, this latter result removes the smallness condition for the initial velocity in a corresponding theorem of R. Danchin and P. B. Mucha.

math.AP↗
arXiv2012

Global wellposedness to incompressible inhomogeneous fluid system with bounded density and non-Lipschitz velocity

In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier-Stokes equations with initial data in critical Besov spaces, which satisfies a non-linear smallness condition. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity field. Furthermore, with additional regularity assumption on the initial velocity or on the initial density, we can also prove the uniqueness of such solution. We should mention that the classical maximal regularity theorem for the heat kernel plays an essential role in this context.

math.AP↗