Partial regularity for steady double phase fluids

Regularity in Sobolev Spaces of Steady Flows of Fluids with Shear-Dependent Viscosity

The system −divS(D(u)) + (u · ∇)u +∇π = f , divu = 0, is considered on a bounded three-dimensional domain under no–stick boundary value conditions, where S has p-structure for some p < 2 and D(u) is the symmetrized gradient of u. Various regularity results for the velocity u and the pressure π in fractional order Sobolev and Nikolskii spaces are obtained.

Partial regularity for quasi minimizers

On Partial Regularity of Steady-state Solutions to the 6d Navier-stokes Equations

Consider steady-state weak solutions to the incompressible Navier-Stokes equations in six spatial dimensions. We prove that the 2D Hausdorff measure of the set of singular points is equal to zero. This problem was mentioned in 1988 by Struwe [24], during his study of the five dimensional case.

Partial Regularity for Flows of H-surfaces

This article studies regularity of weak solutions to the heat equation for H{ surfaces. Under the assumption that the function H is Lipschitz and depends only on the rst two components, the solution has regularity on its domain, except for a set of measure zero. Moreover, if the solution satisses certain energy inequality, this set is nite.

Partial Regularity for Optimal Transport Maps

We prove that, for general cost functions on R, or for the cost d/2 on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.