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Xiangsheng Xu

Publications and source records attributed to Xiangsheng Xu.

32 records · Page 2Linked to original sources
arXiv2020

Regularity results for a nonlinear elliptic-parabolic system with oscillating coefficients

In this paper we study the initial boundary value problem for the system $\mbox{div}(σ(u)\nablaφ)=0,\ \ u_t-Δu=σ(u)|\nablaφ|^2$. This problem is known as the thermistor problem which models the electrical heating of conductors. Our assumptions on $σ(u)$ leave open the possibility that $\liminf_{u\rightarrow\infty}σ(u)=0$, while $\limsup_{u\rightarrow\infty}σ(u)$ is large. This means that $σ(u)$ can oscillate wildly between $0$ and a large positive number as $u\rightarrow \infty$. Thus our degeneracy is fundamentally different from the one that is present in porous medium type of equations. We obtain a weak solution $(u, φ)$ with $|\nabla φ|, |\nabla u|\in L^\infty$ by first establishing a uniform upper bound for $e^{\varepsilon u}$ for some small $\varepsilon$. This leads to an inequality in $\nablaφ$, from whence follows the regularity result. This approach enables us to avoid first proving the Hölder continuity of $φ$ in the space variables, which would have required that the elliptic coefficient $σ(u)$ be an $A_2$ weight. As it is known, the latter implies that $\lnσ(u)$ is "nearly bounded".

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arXiv2020

Global existence of strong solutions to a biological network formulation model in $2+1$ dimensions

In this paper we study the initial boundary value problem for the system\\ $-\mbox{div}\left[(I+\mathbf{m} \mathbf{m}^T)\nabla p\right]=s(x),\ \ \mathbf{m}_t-α^2Δ\mathbf{m}+|\mathbf{m}|^{2(γ-1)}\mathbf{m}=β^2(\mathbf{m}\cdot\nabla p)\nabla p$ in two space dimensions. This problem has been proposed as a continuum model for biological transportation networks. The mathematical challenge is due to the presence of cubic nonlinearities, also known as trilinear forms, in the system. We obtain a weak solution $(\mathbf{m},p) $ with both $|\nabla p|$ and $|\nabla\mathbf{m}|$ being bounded. The result immediately triggers a bootstrap argument which can yield higher regularity for the weak solution. This is achieved by deriving an equation for $v\equiv(I+\mathbf{m} \mathbf{m}^T)\nabla p\cdot\nabla p$, and then suitably applying the De Giorge iteration method to the equation.

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arXiv2020

Global existence of strong solutions to a groundwater flow problem

In this paper we study the initial boundary value problem for the system $Δv= u_{x_1},\ u_t-\mbox{div}\left(\left((a|\mathbf{q}|+m)I+(b-a)\frac{\mathbf{q}\otimes\mathbf{q}}{|\mathbf{q}|}\right)\nabla u\right)=-\nabla u\cdot\mathbf{q}$, where $\mathbf{q}=(-v_{x_2}, v_{x_1})^T$, $\mathbf{q}\otimes\mathbf{q}=\mathbf{q}\mathbf{q}^T$. This problem has been proposed as a model for a fluid flowing through a porous medium under the influence of gravity and hydrodynamic dispersion. For each $T>0$ we obtain a so-called strong solution $(v, u)$ in the function space $L^\infty(0,T; \left(W^{1,\infty}(Ω)\right)^2)$, where $Ω$ is a bounded domain in $\mathbb{R}^2$. The key ingredient in our approach is the decomposition $A^2=\mbox{tr} (A)A-\mbox{det}(A) I$ for any $2\times 2$ symmetric matrix $A$. By exploring this decomposition, we are able to derive an equation of parabolic type for the function $\left(\left((a|\mathbf{q}|+m)I+(b-a)\frac{\mathbf{q}\otimes\mathbf{q}}{|\mathbf{q}|}\right)\nabla u\cdot\nabla u\right)^j, j\geq 1$. With the aid of this equation we obtain a uniform bound for $\nabla u$.

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arXiv2020

Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model

In this paper we first study partial regularity of weak solutions to the initial boundary value problem for the system $-\mbox{div}\left[(I+\mathbf{m}\otimes \mathbf{m})\nabla p\right]=S(x),\ \ \partial_t\mathbf{m}-D^2Δ\mathbf{m}-E^2(\mathbf{m}\cdot\nabla p)\nabla p+|\mathbf{m}|^{2(γ-1)}\mathbf{m}=0$, where $S(x)$ is a given function and $D, E, γ$ are given numbers. This problem has been proposed as a PDE model for biological transportation networks. Mathematically, it seems to have a connection to a conjecture by De Giorgi \cite{DE}. Then we investigate the life-span of classical solutions. Our results show that local existence of a classical solution can always be obtained and the life-span of such a solution can be extended as far away as one wishes as long as the term $\|{\bf m}(x,0)\|_{\infty, Ω}+\|S(x)\|_{\frac{2N}{3}, Ω}$ is made suitably small, where $N$ is the space dimension and $\|\cdot\|_{q,Ω}$ denotes the norm in $L^q(Ω)$.

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arXiv2020

Global existence theorem for a model governing the motion of two cell populations

This article is concerned with the existence of a weak solution to the initial boundary problem for a cross-diffusion system which arises in the study of two cell population growth. The mathematical challenge is due to the fact that the coefficient matrix is non-symmetric and degenerate in the sense that its determinant is $0$. Existence assertion is established by exploring the fact that the total population density satisfies a porous media equation.

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arXiv2019

Special solutions to a fourth-order nonlinear parabolic equation in non-divergence form

In this paper we study a crystal surface model first proposed by H.~Al Hajj Shehadeh, R.V.~Kohn, and J.~Weare (2011 Physica D, 240,1771-1784). By seeking a solution of a particular function form, we are led to a boundary value problem for a fourth-order nonlinear elliptic equation. The mathematical challenge of the problem is due to the fact that the degeneracy in the equation is directly imposed by one of the two boundary conditions. An existence theorem is established in which a meaningful mathematical interpretation of one of the boundary conditions remains open. Our proof seems to suggest that this is unavoidable. We also obtain self-similar solutions to the crystal surface model which are positive and unbounded. This is in sharp contrast with the linear biharmonic heat equation.

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arXiv2018

Existence theorems for a crystal surface model involving the p-Laplace operator

The manufacturing of crystal films lies at the heart of modern nanotechnology. How to accurately predict the motion of a crystal surface is of fundamental importance. Many continuum models have been developed for this purpose, including a number of PDE models, which are often obtained as the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. In this paper we offer an analytical perspective into some of these models. To be specific, we study the existence of a weak solution to the boundary value problem for the equation $ - Δe^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right)}+au=f$, where $p>1, a>0$ are given numbers and $f$ is a given function. This problem is derived from a crystal surface model proposed by J.L.~Marzuola and J.~Weare (2013 Physical Review, E 88, 032403). The mathematical challenge is due to the fact that the principal term in our equation is an exponential function of a p-Laplacian. Existence of a suitably-defined weak solution is established under the assumptions that $p\in(1,2], \ N\leq 4$, and $f\in W^{1,p}$. Our investigations reveal that the key to our existence assertion is how to control the set where $-\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right)$ is $\pm\infty$.

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arXiv2018

Logarithmic upper bounds for weak solutions to a class of parabolic equations

It is well known that a weak solution $φ$ to the initial boundary value problem for the uniformly parabolic equation $\partial_tφ-\mbox{div}(A\nabla φ) +ωφ= f $ in $Ω_T\equivΩ\times(0,T)$ satisfies the uniform estimate $$ \|φ\|_{\infty,Ω_T}\leq \|φ\|_{\infty,\partial_pΩ_T}+c\|f\|_{q,Ω_T}, \ \ \ c=c(N,λ, q, Ω_T), $$ provided that $q>1+\frac{N}{2}$, where $Ω$ is a bounded domain in $\mathbb{R}^N$ with Lipschitz boundary, $T>0$, $\partial_pΩ_T$ is the parabolic boundary of $Ω_T$, $ω\in L^1(Ω_T)$ with $ω\geq 0$, and $λ$ is the smallest eigenvalue of the coefficient matrix $A$. This estimate is sharp in the sense that it generally fails if $q=1+\frac{N}{2}$. In this paper we show that the linear growth of this upper bound in $\|f\|_{q,Ω_T}$ can be improved. To be precise, we establish \begin{equation*} \|φ\|_{\infty,Ω_T}\leq \|φ_0\|_{\infty,\partial_pΩ_T}+c\|f\|_{1+\frac{N}{2},Ω_T}\left(\ln(\|f\|_{q,Ω_T}+1)+1\right). \end{equation*}

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arXiv2018

Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface

In this work we consider $$ w_t=[(w_{hh}+c_0)^{-3}]_{hh},\qquad w(0)=w^0, $$ which is derived from a thin film equation for epitaxial growth on vicinal surface. We formulate the problem as the gradient flow of a suitably-defined convex functional in a non-reflexive space. Then by restricting it to a Hilbert space and proving the uniqueness of its sub-differential, we can apply the classical maximal monotone operator theory. The mathematical difficulty is due to the fact that $w_{hh}$ can appear as a positive Radon measure. We prove the existence of a global strong solution. In particular, the equation holds almost everywhere when $w_{hh}$ is replaced by its absolutely continuous part.

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arXiv2018

Partial regularity of weak solutions to a PDE system with cubic nonlinearity

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.

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arXiv2017

Local and global existence of solutions to a fourth-order parabolic equation modeling kinetic roughening and coarsening in thin films

In this paper we study both the Cauchy problem and the initial boundary value problem for the equation $\partial_tu+\mbox{div}\left(\nablaΔu-{\bf g}(\nabla u)\right)=0$. This equation has been proposed as a continuum model for kinetic roughening and coarsening in thin films. In the Cauchy problem, we obtain that local existence of a weak solution is guaranteed as long as the vector-valued function ${\bf g}$ is continuous and the initial datum $u_0$ lies in $C^1(\mathbb{R}^N)$ with $\nabla u_0(x)$ being uniformly continuous and bounded on $\mathbb{R}^N$ and that the global existence assertion also holds true if we assume that ${\bf g}$ is locally Lipschitz and satisfies the growth condition $|{\bf g}(ξ) |\leq c|ξ|^α$ for some $c>0, α\in (2, 3)$, $\sup_{\mathbb{R}^N}|\nabla u_0|<\infty$, and the norm of $u_0$ in the space $L^{\frac{(α-1)N}{3-α}}(\mathbb{R}^N) $ is sufficiently small. This is done by exploring various properties of the biharmonic heat kernel. In the initial boundary value problem, we assume that ${\bf g}$ is continuous and satisfies the growth condition $|{\bf g}(ξ) |\leq c|ξ|^α+c$ for some $c, α\in (0,\infty)$. Our investigations reveal that if $α\leq 1$ we have global existence of a weak solution, while if $1<α<\frac{N^2+2N+4}{N^2}$ only a local existence theorem can be established. Our method here is based upon a new interpolation inequality, which may be of interest in its own right.

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arXiv2017

Regularity theorems for a biological network formulation model in two space dimensions

We present several regularity results for a biological network formulation model originally introduced by D. Cai and D. Hu in {\it Phys. Rev. Lett.}, {\bf 111}(2013), 138701. A consequence of these result is that a stationary weak solution must be a classical one in two space dimensions. Our mathematical analysis is based upon the weakly monotone function theory and Hardy space methods.

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arXiv2017

Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions

In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation $\frac{\partial u}{\partial t} = Δ\left(Δu\right)^{-3}$. This problem arises in the mathematical modeling of the evolution of a crystal surface. Existence of a weak solution $u$ with $Δu\geq 0$ is obtained via a suitable substitution. Our investigations reveal the close connection between this problem and the equation $\partial_tρ+ρ^2Δ^2ρ^3=0$, another crystal surface model first proposed by H. Al Hajj Shehadeh, R. V. Kohn and J. Weare in Physica D: Nonlinear Phenomena, bf 240 (2011), no. 21, 1771-1784.

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