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

2023/08/22 by Le, Dung
#35B65 #35J70 #42B37 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2308.11678

Abstract

We establish several bounds for solutions to elliptic/parabolic cross-diffusion systems of m equations (m≥2) 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 \RR³ 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).

Related

• Regularity of Solutions to a Class of Degenerate Cross Diffusion Systems of m Equations
• Global Existence and Regularity Results for Large Cross Diffusion Models on Planar Domains
• Global Existence to a Class of Triangular Block Matrix Cross Diffusion Systems and the Spectral Gap Condition
• Perturbative global solutions of a large class of cross diffusion systems in any dimension
• On the Global Existence of a Class of Strongly Coupled Parabolic Systems