A path in a Riemannian manifold can be approximated by a path meeting only finitely many times the cut locus of a given point. The proof of this property uses recent works of Itoh–Tanaka and Li–Nirenberg about the differential structure of the cut locus. We present applications in the regularity theory of optimal transport.

We introduce a concept of viscosity solutions for Hamilton-Jacobi equations (HJE) in the Wasserstein space. We prove existence of solutions for the Cauchy problem for certain Hamiltonians defined on the Wasserstein space over the real line. In order to illustrate the link between HJE in the Wasserstein space and Fluid Mechanics, in the last part of the paper we focus on a special Hamiltonian. The characteristics for these HJE are solutions of physical systems in finite dimensional spaces.

For a semilinear heat equation admitting blow-up solutions a sufficient condition for nonexistence of local-in-time solutions are obtained. In particular, it is shown that if an initial data tends to infinity at space infinity then there is no local-in-time solution. As an application if the solution blows up at space infinity with least blow-up time, the solution cannot be extendable after blow-up time.

We study Maxwell’s equations in a quasi-static electromagnetic field, where the electrical conductivity of the material depends on the temperature. By establishing the reverse Hölder inequality, we prove partial regularity of weak solutions to the non-linear elliptic system and the non-linear parabolic system in a quasi-static electromagnetic field.

We investigate the large-time behavior of solutions of the Cauchy problem for Hamilton-Jacobi equations on the real line $R$. We establish a result on convergence of the solutions to asymptotic solutions as time $t$ goes to infinity.

In this paper, we prove that for any strictly convex polynomial, or more generally any strictly convex function satisfying appropriate conditions, there is an associated Sobolev type inequality.