In this talk, we study continuous maximal regularity theory for a class of degenerate or singular differential operators on manifolds with singularities. Based on this theory, we show local existence and uniqueness of solutions for several nonlinear geometric flows and diffusion equations on non-compact, or even incomplete, manifolds, including the Yamabe flow and parabolic p-Laplacian equations. In addition, we also establish regularity properties of solutions by means of a technique consisting of continuous maximal regularity theory, a parameter-dependent diffeomorphism and the implicit function theorem.
Understanding the incompressible/compressible fluid is a fundamental, but
challenging, project not only in numerical analysis, but also in
theoretical analysis, especially when extra effects, such as the elastic
deformation or the magnetic field, interact with the flow. In this talk,
the incompressible fluid and its associated flow map will be reviewed first.
The main object of this talk devotes to a recent work in understanding
incompressible/compressible magnetohydrodynamic fluids with zero magnetic
diffusivity (which is equivalent to infinite conductivity). This is a
joint work with Fanghua Lin.
We study the effect of defects in the periodic homogenization of
Hamilton-Jacobi equations with non convex Hamiltonians. More precisely, we
handle the question about existence of sublinear solutions of the cell
In this talk, I will discuss recent results on the
large time well-posedness of classical solutions to the
multi-dimensional compressible Navier-Stokes system with possible
large oscillations and vacuum.
The focus will be on finite-time blow-up of classical solutions for
the 3-D full compressible Navier-Stokes system, and the global
existence of classical solutions to the isentropic compressible
Navier-Stokes system in both 2-D and 3-D in the presence of vacuum
and possible large oscillations. New estimates on the fast decay
of the pressure in the presence of vacuum will be presented, which
are crucial for the well-posedness theory in 2-dimensional case.
In this talk we will present various results on the size of the nodal (zero) set for solutions of partial differential equations of elliptic and parabolic type. In particular, we will establish a sharp upper bound for the (n-1)-dimensional Hausdorff measure of the nodal sets of the eigenfunctions of regular analytic elliptic problems. We will also show certain more recent results concerning semilinear equations (e.g. Navier-Stokes equations) and equations with non-analytic coefficients.