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Jishan Fan

Publications and source records attributed to Jishan Fan.

9 recordsLinked to original sources
arXiv2026

Conditonal Lipschitz stability for the Inverse Problem of the 2D Navier-Stokes System in a Bounded Domain

This paper concerns an inverse problem for the initial boundary value problem of the two-dimensional Navier-Stokes system defined in a bounded simply connected domain with slip, vorticity boundary conditions, and a global vorticity invariant constraint. We establish conditional Lipschitz stability and a local recovery for this inverse problem, where the velocity field and space-independent boundary vorticity are locally recovered from the given initial velocity field and the global vorticity invariant. Our analysis is based on well-posedness estimates and energy methods for the vorticity transport equation.

math.AP↗
arXiv2020

Short-time approximate solutions of an equation modeling a camphor motion

As a profound example of spontaneous motion, we analyze the motion of a camphor particle on a water surface. The motion is modeled as an initial-boundary value problem for a coupled nonlinear system of a diffusion equation and an ordinary differential equation in a two-dimensional domain. Since it seems that the well-posedness of this initial boundary value problem is missing, we provided its proof. Then, by constructing an approximate solution to this initial boundary value problem, we gave a mathematically rigorous interpretation of a camphor motion. That is we showed that the motion of camphor locally in time has a self-avoiding orbit. We also gave the numerical performance of the approximate solution.

math.AP↗
arXiv2017

Global strong solutions to the 3D full compressible Navier-Stokes system with vacuum in a bounded domain

In this short paper we establish the global well-posedness of strong solutions to the 3D full compressible Navier-Stokes system with vacuum in a bounded domain $Ω\subset \mathbb{R}^3$ by the bootstrap argument provided that the viscosity coefficients $λ$ and $μ$ satisfy that $7λ>9μ$ and the initial data $ρ_0$ and $u_0$ satisfy that $\|ρ_0\|_{L^\infty(Ω)}$ and $\|ρ_0|u_0|^5\|_{L^1(Ω)}$ are sufficient small.

math.AP↗
arXiv2017

Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vaccum

This paper considers the initial boundary problem to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. The global existence and uniqueness of large strong solutions are established when the heat conductivity coefficient $κ(θ)$ satisfies \begin{equation*} C_{1}(1+θ^q)\leq κ(θ)\leq C_2(1+θ^q) \end{equation*} for some constants $q>0$, and $C_1,C_2>0$.

math.AP↗
arXiv2015

Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain

In this paper we establish the uniform estimates of strong solutions with respect to the Mach number and the dielectric constant to the full compressible Navier-Stokes-Maxwell system in a bounded domain. Based on these uniform estimates, we obtain the convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations for well-prepared data.

math.AP↗
arXiv2011

A gradient estimate for solutions to parabolic equations with discontinuous coefficients

Li-Vogelius and Li-Nirenberg gave a gradient estimate for solutions of strongly elliptic equations and systems of divergence forms with piecewise smooth coefficients, respectively. The discontinuities of the coefficients are assumed to be given by manifolds of codimension 1, which we called them manifolds of discontinuities. Their gradient estimate is independent of the distances between manifolds of discontinuities. In this paper, we gave a parabolic version of their results. That is, we gave a gradient estimate for parabolic equations of divergence forms with piecewise smooth coefficients. The coefficients are assumed to be independent of time and their discontinuities are likewise the previous elliptic equations. As an application of this estimate, we also gave a pointwise gradient estimate for the fundamental solution of a parabolic operator with piecewise smooth coefficients. The both gradient estimates are independent of the distances between manifolds of discontinuities.

math.AP↗