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 …
arises in the study of biological transport networks. The system consists of a possibly …
A local partial regularity theorem for weak solutions of degenerate elliptic equations and its application to the thermistor problem
X Xu - 1999 - projecteuclid.org
A partial regularity theorem is established for weak solutions of elliptic equations of the form
div(A(y)∇ψ)=0. Here we allow the possibility that the eigenvalues of A(y) are not bounded …
div(A(y)∇ψ)=0. Here we allow the possibility that the eigenvalues of A(y) are not bounded …
Global existence of strong solutions to a biological network formulation model in dimensions
X Xu - arXiv preprint arXiv:1911.01970, 2019 - arxiv.org
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-\alpha^ 2\Delta\mathbf …
[(I+\mathbf {m}\mathbf {m}^ T)\nabla p\right]= s (x),\\\mathbf {m} _t-\alpha^ 2\Delta\mathbf …
Hölder continuity of weak solutions to an elliptic-parabolic system modeling biological transportation network
X Xu - Journal of Differential Equations, 2023 - Elsevier
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 …
modeling natural network formation. The system is singular and involves cubic nonlinearity …
Partial regularity of weak solutions and life-span of smooth solutions to a biological network formulation model
X Xu - SN Partial Differential Equations and Applications, 2020 - Springer
In this paper we study partial regularity of weak solutions to the initial boundary value
problem for the system-div\left (I+ m ⊗ m) ∇ p\right= S (x),\∂ _t mD^ 2 Δ mE^ 2 (m ⋅ ∇ p) ∇ …
problem for the system-div\left (I+ m ⊗ m) ∇ p\right= S (x),\∂ _t mD^ 2 Δ mE^ 2 (m ⋅ ∇ p) ∇ …
Uniform BMO estimate of parabolic equations and global well-posedness of the thermistor problem
B Li, C Yang - Forum of Mathematics, Sigma, 2015 - cambridge.org
We prove global well-posedness of the time-dependent degenerate thermistor problem by
establishing a uniform-in-time bounded mean ocsillation (BMO) estimate of inhomogeneous …
establishing a uniform-in-time bounded mean ocsillation (BMO) estimate of inhomogeneous …
Local partial regularity theorems for suitable weak solutions of a class of degenerate systems
X Xu - Applied Mathematics and Optimization, 1996 - Springer
A partial regularity theorem is established for a particular class of weak solutions to the
system∂ u/∂ t− div (K (u)▽ u)= σ (u)¦▽ ϕ¦ 2, div (σ (u)▽ ϕ)= 0 on a bounded domain in RN …
system∂ u/∂ t− div (K (u)▽ u)= σ (u)¦▽ ϕ¦ 2, div (σ (u)▽ ϕ)= 0 on a bounded domain in RN …
Mathematical validation of a continuum model for relaxation of interacting steps in crystal surfaces in 2 space dimensions
X Xu - Calculus of Variations and Partial Differential …, 2020 - Springer
In this paper we study the boundary value problem for the equation div\left (D (∇ u) ∇\left
(div\left (| ∇ u|^ p-2 ∇ u+ β ∇ u| ∇ u|\right)\right)\right)+ au= f div D (∇ u)∇ div|∇ u| p-2∇ …
(div\left (| ∇ u|^ p-2 ∇ u+ β ∇ u| ∇ u|\right)\right)\right)+ au= f div D (∇ u)∇ div|∇ u| p-2∇ …
Local regularity theorems for the stationary thermistor problem
X Xu - Proceedings of the Royal Society of Edinburgh Section …, 2004 - cambridge.org
In this paper we present a simpler proof of a result of Lewis concerning the continuity of
weak solutions to the two-dimensional thermistor problem in the case where the temperature …
weak solutions to the two-dimensional thermistor problem in the case where the temperature …