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Dung Le

Publications and source records attributed to Dung Le.

At least 37 records · Page 2Linked to original sources
arXiv2023

Global Existence to a Class of Triangular Block Matrix Cross Diffusion Systems and the Spectral Gap Condition

We study the global existence of classical solutions to cross diffusion systems of $m$ equations on $N$-dimensional domains ($m,N\ge2$). The diffusion matrix is a triangular block matrix with coupled entries. We establish that the $W^{1,p}$ norm of solutions for some $p>N$ does not blow up in finite time so that the results in \cite{Am2} is applicable. We will also show that the spectral gap condition in \cite{dlebook,dlebook1} can be relaxed via a new result on BMO norms in \cite{dleBMO}.

math.AP↗
arXiv2023

A priori Bound of Solutions to a Class of Elliptic/Parabolic Cross Diffusion Systems of $m$ Equations on Two/Three Dimensional Domains. Existence on thin domains

We establish several bounds for solutions to elliptic/parabolic cross-diffusion systems of $m$ equations ($m\ge2$) on 2d/3d domains $\Og$. We settle the existence and global existence problems in these cases and also provide new counter-examples for the case of general dimensions. Most importantly, we prove that when $m=N=3$, the thinness of $\Og$ in $\RR^3$ is sufficient and necessary. When $m$ is arbitrary and $N=3$, we establish global existence results for nonlinear cross-diffusion systems (the case of the scalar semilinear equation was considered in the literature but the classical methods do not seem to be applicable here).

math.AP↗
arXiv2023

Strong Persistence of a Class of Strongly Coupled Parabolic Systems of $m$ Equations

We establish one of the most important assumptions of the strong persistence theory for dynamical systems associated to cross diffusion systems of $m$ equations ($m\ge2$): the stable sets of semi-trivial steady cannot intersect the interior of the positive cone of $C(Ω,\mathbb{R}^m)$. Many examples will be provided to show the effects of the cross diffusion.

math.AP↗
arXiv2023

Some Maximum Principles for Cross Diffusion Systems

We establish certain maximum principles for a class of strongly coupled elliptic (or cross diffusion) systems of $m\ge2$ equations. The reaction parts can be non cooperative. These new results will be crucial in obtaining coexistence and persistence for many models with cross diffusion effects.

math.AP↗
arXiv2022

Fast Approximation of the Generalized Sliced-Wasserstein Distance

Generalized sliced Wasserstein distance is a variant of sliced Wasserstein distance that exploits the power of non-linear projection through a given defining function to better capture the complex structures of the probability distributions. Similar to sliced Wasserstein distance, generalized sliced Wasserstein is defined as an expectation over random projections which can be approximated by the Monte Carlo method. However, the complexity of that approximation can be expensive in high-dimensional settings. To that end, we propose to form deterministic and fast approximations of the generalized sliced Wasserstein distance by using the concentration of random projections when the defining functions are polynomial function, circular function, and neural network type function. Our approximations hinge upon an important result that one-dimensional projections of a high-dimensional random vector are approximately Gaussian.

stat.ML↗
arXiv2022

Entropic Gromov-Wasserstein between Gaussian Distributions

We study the entropic Gromov-Wasserstein and its unbalanced version between (unbalanced) Gaussian distributions with different dimensions. When the metric is the inner product, which we refer to as inner product Gromov-Wasserstein (IGW), we demonstrate that the optimal transportation plans of entropic IGW and its unbalanced variant are (unbalanced) Gaussian distributions. Via an application of von Neumann's trace inequality, we obtain closed-form expressions for the entropic IGW between these Gaussian distributions. Finally, we consider an entropic inner product Gromov-Wasserstein barycenter of multiple Gaussian distributions. We prove that the barycenter is a Gaussian distribution when the entropic regularization parameter is small. We further derive a closed-form expression for the covariance matrix of the barycenter.

math.ST↗
arXiv2021

On the Global Existence of a Class of Strongly Coupled Parabolic Systems

We establish the existence of strong solutions to a class of cross diffusion systems on $\RR^N$ consists of $m$ equations ($m,N\ge 2$). which generalizes the Shigesada-Kawasaki-Teramoto (SKT) model in population dynamics. We introduce the concept of a {\em strong-weak solution} of the systems and show that their existence can be established under weaker conditions. These {\em strong-weak solutions} coincide with strong solutions so that the existence of strong solutions is proved. The SKT model on planar domains ($N=2$) with cubic diffusions and advections is completely solved.

math.AP↗
arXiv2021

On the Global Existence of a Class of Strongly Coupled Parabolic Systems

We establish the global existence of a class of strongly coupled parabolic systems. The necessary apriori estimates will be obtained via our new approach to the regularity theory of parabolic scalar equations with integrable data and new $W^{1,p}$ estimates of their solutions. The key assumption here is that the $L^p$ norms of solutions are uniformly bounded for some sufficiently large $p\in (1,\infty)$, an assumption can be easily affirmed for systems with polynomial growth data. This replaces the usual condition that the solutions are uniformly bounded which is very hard to be verified because maximum principles for systems are generally unavailable.

math.AP↗
arXiv2020

On the Positivity of Weak Solutions to a Class of Cross Diffusion Systems

We establish the positivity of weak (and very weak) solutions to a class of cross diffusion systems which is inspired by models in mathematical biology/ecology, in particular the Shigesada-Kawasaki-Teramoto (SKT) model in population biology. Examples and counterexamples will be provided to show that our contions are near optimal.

math.AP↗
arXiv2019

Global Existence for some Cross Diffusion Systems with Equal Cross Diffusion/Reaction Rates

We consider some cross diffusion systems which is inspired by models in mathematical biology/ecology, in particular the Shigesada-Kawasaki-Teramoto (SKT) model in population biology. We establish the global existence of strong solutions to systems for multiple species having equal either diffusion or reaction rates. The systems are given on bounded domains of arbitrary dimension.

math.AP↗
arXiv2019

Existence and Uniqueness of Weak Solutions to a Class of Degenerate Cross Diffusion Systems

We consider a class of cross diffusion systems with degenerate (or porous media type) diffusion which is inspired by models in mathematical biology/ecology with zero self diffusions. Known techniques for scalar equations are no longer available here as maximum/comparison principles are generally unavailable for systems. However, we will provide the existence of weak solutions to the degenerate systems under mild integrability conditions of strong solutions to nondegenerate systems and show that they converge to a weak solution of the degerate system. These conditions will be verified for the model introduced by Shigesada {\it et al.}. Uniqueness of limiting and unbounded weak solutions will also be proved.

math.AP↗
arXiv2019

Uniqueness and Regularity of Unbounded Weak Solutions to a Class of Cross Diffusion Systems

We establish the uniqueness and regularity of weak (and very weak) solutions to a class of cross diffusion systems which is inspired by models in mathematical biology/ecology, in particular the Shigesada-Kawasaki-Teramoto (SKT) model in population biology. No boundedness assumption on these solutions is supposed here as known techniques for scalar equations such as maximum/comparison principles are generally unavailable for systems. Furthermore, for planar domains we show that unbounded weak solutions satisfying mild integrability conditions are in fact smooth.

math.AP↗
arXiv2017

On the Solvability of a Class of Degenerate or Singular Strongly Coupled Parabolic Systems

The existence of strong solutions to general class of strongly coupled parabolic systems will be discussed. These systems can be degenerate or singular as boundedness of theirs solutions are unavailable and not assummed. The results greatly improve those in a recent papers \cite{letrans,dleJFA,dleANS} as the systems can have quadratic growth in gradients. A unified proof for both cases is presented. Most importantly, the VMO assumption in \cite{dleJFA,dleANS} will be replaced by a much versatile one thanks to a new local weighted Gagliardo-Nirenberg involving BMO norms. Degenerate and singular generalized SKT models in biology will be presented as a nontrivial application of the main theorem.

math.AP↗
arXiv2017

Existence of Strong Solutions to Degenerate or Singular Strongly Coupled Elliptic Systems

A general class of strongly coupled elliptic systems with quadratic growth in gradients is considered and the existence of their strong solutions is established. The results greatly improve those in a recent paper \cite{dleJFA} as the systems can be either degenerate or singular when their solutions become unbounded. A unified proof for both cases is presented. Most importantly, the VMO assumption in \cite{dleJFA} will be replaced by a much versatile one thanks to a new local weighted Gagliardo-Nirenberg involving BMO norms. Examples in physical models will be provided.

math.AP↗
arXiv2016

Weighted Gagliardo-Nirenberg Inequalities Involving BMO Norms and Measures

Global and local weighted Gagliardo-Nirenberg inequalities with doubling measures are established. These inequalities are key ingredients for the regularity theory and existence of strong solutions for strongly coupled parabolic and elliptic systems which are degenerate or singular because of the unboundedness of dependent and independent variables.

math.FA↗
arXiv2016

Local and Global Existence of Strong Solutions to Large Cross Diffusion Systems

We study the solvability of a general class of cross diffusion systems and establish the local and global existence of their strong solutions under the weakest assumption that they are VMO. This work simplifies the setting in our previous work [16] and provides new extensions which are more verifiable in applications.

math.AP↗