Special Session 189: Analysis and applications of elliptic and parabolic equations

Partial Regularity for Navier-Stokes equations

Lihe Wang University of Iowa USA Co-Author(s):    Lihe Wang

Abstract: We will present a new compactness lemma that can handle the noncompactness in the present and through which proves the partial regularity theorem.

Regularity for $\omega$-minimizers of general $p$-Laplacian type functionals with matrix weights

Rui Yang Central South University Peoples Rep of China Co-Author(s):

Abstract: We study general $p$-Laplacian type functionals with matrix weights, which may cause degeneracy, singularity, or both. We establish an optimal Calderon-Zygmund theory for any $\omega$-minimizer of such weighted energy functionals under a smallness log-BMO condition on the matrix weight and quantitative control of $\omega$-minimality, the weighted gradient enjoys the same integrability as the weighted nonhomogeneous term. This is joint work with Sun-sig Byun.

Mathematical analysis of a tumor invasion model

Shulin Zhou Peking University Peoples Rep of China Co-Author(s):

Abstract: In this talk I will introduce a reaction-diffusion system modelling tumor invasion. The underlying partial differential equation system has cross-diffusion and density-dependent diffusion coefficient and involves the interactions of three quantities: tumor tissue density, acid concentration and normal tissue density. The most distinguished feature of this model is that the diffusion of tumor cells is influenced by the density of normal cells and diffusion degeneracy arises when normal cells are at their carrying capacity. Therefore, this system is strongly coupled and might be degenerate. I will review the known mathematical results of this system and present our new results of this system.