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Xiangsheng Xu

Publications and source records attributed to Xiangsheng Xu.

At least 19 recordsLinked to original sources
arXiv2026

Large data existence of global-in-time strong solutions to the incompressible Navier-Stokes equations in high space dimensions

We study the existence of a strong solution to the initial value problem for the incompressible Navier-Stokes equations in the whole space. Our investigation shows that a ``suitable'' weak solution to the problem becomes a strong one whenever the initial velocity is divergence free and uniformly bounded with finite energy. Our results seem to have given a positive answer to the Navier-Stokes millennium problem proposed by the Clay Mathematical Institute.

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arXiv2026

Global existence of a strong solution to the initial value problem for the Nernst-Planck-Navier-Stokes system in high space dimensions

We study the existence of a strong solution to the initial value problem for the Nernst-Planck-Navier-Stokes (NPNS) system in $\mathbb{R}^N, N\geq 3$. The system describes the electrodiffusion of ions in a viscous Newtonian fluid. A strong solution is obtained in any dimension of space without constraints on the number of species or the size of the given data.

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arXiv2025

A Global Existence Theorem for a Fourth-Order Crystal Surface Model with Gradient Dependent Mobility

In this article we study the existence of solutions to a fourth-order nonlinear PDE related to crystal surface growth. The key difficulty in the equations comes from the mobility matrix, which depends on the gradient of the solution. When the mobility matrix is the identity matrix there are now many existence results, however when it is allowed to depend on the solution we lose crucial estimates in the time direction. In this work we are able to prove the global existence of weak solutions despite this lack of estimates in the time direction.

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arXiv2024

An Existence Theorem for a Model of Temperature Within a Lithium-Ion Battery

In this article we investigate a model for the temperature within a Lithium-Ion battery. The model takes the form of a parabolic PDE for the temperature coupled with two elliptic PDE's for the electric potential within the solid and electrolyte phases. The primary difficulty comes from the coupling term, which is given by the Butler-Volmer equation. It features an exponential nonlinearity of both the electric potentials and the reciprocal of the temperature. Another difficulty arising in the temperature equation are the gradients of the electric potentials squared showing up on the right-hand side. Due to the nonlinearity, meaningful estimates for the temperature are currently not known. In spite of this, our investigation reveals the local existence of continuous temperature for the Lithium-Ion Battery.

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arXiv2023

Modulus of continuity of weak solutions to a class of singular elliptic equations

In this paper we study the modulus of continuity of weak solutions to a singular elliptic equation in the plane under very weak assumption on the integrability of the elliptic coefficients. Our investigation reveals that the modulus of continuity can be described by the reciprocal of the logarithmic function raised to a power. However, the power can be arbitrarily large. This is in sharp contrast with a result by J. Onninen and X. Zhong for a degenerate elliptic equation in the plane, in which the power must be suitably small.

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arXiv2023

Existence theorem for a partially parabolic cross-diffusion system

We study an initial boundary value problem for a cross-diffusion system in population dynamics. The mathematical challenge is due to the fact that the determinant of the coefficient matrix of the system changes signs. As a result, the system is only partially parabolic. We design an approximation scheme. The sequence of approximate solutions generated by our scheme converges and its limit satisfies the original system in the parabolic region. It remains open if one can construct a vector-valued function that satisfies the system in both the parabolic region and the anti-parabolic one.

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arXiv2023

Exponential Crystal Relaxation Model With P-Laplacian

In this article we prove the global existence of weak solutions to an initial boundary value problem with an exponential and p-Laplacian nonlinearity. The equation is a continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation. In our investigation we find a weak solution where the exponent in the equation, $-Δ_p u$, can have a singular part in accordance with the Lebesgue Decomposition Theorem. The singular portion of $-Δ_p u$ corresponds to where $-Δ_p u = -\infty$, which leads it to have a canceling effect with the exponential nonlinearity. This effect has already been demonstrated for the case of a linear exponent $p=2$, and for the time independent problem. Our investigation reveals that we can exploit this same effect in the time dependent case with nonlinear exponent. We obtain a solution by first forming a sequence of approximate solutions and then passing to the limit. The key to our existence result lies in the observation that one can still obtain the precompactness of the term $e^{-Δ_p u}$ despite a complete lack of estimates in the time direction. However, we must assume that $1<p\leq 2$.

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arXiv2022

Hölder continuity of weak solutions to an elliptic-parabolic system modeling biological transportation network

In this paper we study the regularity of weak solutions to an elliptic-parabolic system modeling natural network formation. The system is singular and involves cubic nonlinearity. Our investigation reveals that weak solutions are Hölder continuous when the space dimension $N$ is $2$. This is achieved via an inequality associated with the Stummel-Kato class of functions and refinement of a lemma originally due to S. Campanato and C. B. Morrey (\cite{G}, p. 86).

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arXiv2022

Existence Theorems for a Fourth-Order Exponential PDE Related to Crystal Surface Growth

In this article we prove the global existence of a unique strong solution to the initial boundary-value problem for a fourth-order exponential PDE. The equation we study was originally proposed to study the evolution of crystal surfaces, and was derived by applying a nonstandard scaling regime to a microscopic Markov jump process with Metropolis rates. Our investigation here finds that compared to the PDE's which use Arhenious rates, (and also have a fourth order exponential nonlinearity) the hyperbolic sine nonlinearity in our equation can offer much better control over the exponent term even in high dimensions.

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arXiv2022

Partial regularity for an exponential PDE in crystal surface models

We study the regularity properties of a weak solution to the boundary value problem for the equation $-Δρ+a u=f$ in a bounded domain $Ω\subset \mathbb{R}^N$, where $ρ=e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla u+β_0|\nabla u|^{-1}\nabla u\right)}$. This problem is derived from the mathematical modeling of crystal surfaces. It is known that the exponent term can exhibit singularity. In this paper we obtain a partial regularity result for the weak solution. It asserts that there exists an open subset $Ω_0\subset Ω$ such that $|Ω\setminusΩ_0|=0$ and the exponent term is locally bounded in $Ω_0$. Furthermore, if $x_0\in Ω\setminusΩ_0$, then $ρ$ vanishes of $N+2-\varepsilon$ order at $x_0$ for each $\varepsilon\in(0,2)$. Our results reveal that the exponent term behaves well if it stays away from negative infinity.

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arXiv2022

Life span of solutions to a PDE model for Lithium-ion batteries in high space dimensions

In this paper we study a system of partial differential equations which models lithium-ion batteries. The system describes the conservation of Lithium and conservation of charges in the solid and electrolyte phases, together with the conservation of energy. The mathematical challenge is due to the fact that the reaction terms in the system involve the hyperbolic sine function along with possible degeneracy in one of the high order terms. We obtain a local existence assertion for the initial boundary problem for the system which offers insight into how long a battery can last.

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arXiv2021

Strong solutions to a fourth order exponential PDE describing epitaxial growth

In this paper we prove the global existence of a strong solution to the initial boundary value problem for the exponential partial differential equation $\partial_tu-Δe^{-Δu}+e^{-Δu}-1=0$. The equation was proposed as a continuum model for epitaxial growth of crystal surfaces on vicinal surfaces with evaporation and deposition effects \cite{GLLM}. Our investigations reveal that we must control the size of both $\left\| e^{-Δu(x,0)}\right\|_{W^{2,2}(Ω)}$ and $\left\| e^{Δu(x,0)}\right\|_{\infty,Ω}$ suitably to achieve our results. Related results in \cite{GM,LS} were established via the Weiner algebra framework. Here we offer a totally new approach, which seems to shed more light on the nature of exponential nonlinearity.

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arXiv2021

Blow-up time of strong solutions to a biological network formation model in high space dimensions

We investigate the possible blow-up of strong solutions to a biological network formation model originally introduced by D. Cai and D. Hu \cite{HC}. The model is represented by an initial boundary value problem for an elliptic-parabolic system with cubic non linearity. We obtain an algebraic equation for the possible blow-up time of strong solutions. The equation yields information on how various given data may contribute to the blow-up of solutions. As a by-product of our development, we establish a $W^{1,q}$ estimate for solutions to an elliptic equation which shows the explicit dependence of the upper bound on the elliptic coefficients.

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arXiv2021

Existence and incompressible limit of a tissue growth model with autophagy

In this paper we study a cross-diffusion system whose coefficient matrix is non-symmetric and degenerate. The system arises in the study of tissue growth with autophagy. The existence of a weak solution is established. We also investigate the limiting behavior of solutions as the pressure gets stiff. The so-called incompressible limit is a free boundary problem of Hele-Shaw type. Our key new discovery is that the usual energy estimate still holds as long as the time variable stays away from $0$.

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arXiv2020

Nonlinear diffusion in the Keller-Segel model of parabolic-parabolic type

In this paper we study the initial boundary value problem for the system $u_t-Δu^m=-\mbox{div}(u^{q}\nabla v),\ v_t-Δv+v=u$. This problem is the so-called Keller-Segel model with nonlinear diffusion. Our investigation reveals that nonlinear diffusion can prevent overcrowding. To be precise, we show that solutions are bounded as long as $m>q>0$, thereby substantially generalizing the known results in this area. Furthermore, our result seems to imply that the Keller-Segel model can have bounded solutions and blow-up ones simultaneously.

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arXiv2020

Mathematical validation of a continuum model for relaxation of interacting steps in crystal surfaces in $2$ space dimensions

In this paper we study the boundary value problem for the equation $\mbox{div}\left(D(\nabla u)\nabla\left(\mbox{div}\left(|\nabla u|^{p-2}\nabla u+β\frac{\nabla u}{|\nabla u|}\right)\right)\right)+au=f$ in the $z=(x,y)$ plane. This problem is derived from a continuum model for the relaxation of a crystal surface below the roughing temperature. The mathematical challenge is of two folds. First, the mobility $D(\nabla u)$ is a $2\times 2$ matrix whose smallest eigenvalue is not bounded away from $0$ below. Second, the equation contains the $1$-Laplace operator, whose mathematical properties are still not well-understood. Existence of a weak solution is obtained. In particular, $|\nabla u|$ is shown to be bounded when $p>\frac{4}{3}$.

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