Abstract: We consider the nonlinear instability of a steady state
$v_*$ of the Euler equation in an $n$-dim fixed smooth bounded domain. When
considered in $H^s$, $s>1$, at the linear level, the stretching of the
steady fluid trajectories induces unstable essential spectrum which
corresponds to linear instability at small spatial scales and the
corresponding growth rate depends on the choice of the space $H^s$.
More physically interesting linear instability relies on the unstable
eigenvalues which correspond to large spatial scales. In the case when
the linearized Euler equation at $v_*$ has an exponential dichotomy of
center-stable and unstable (from eigenvalues) directions, most of the
previous results obtaining the expected nonlinear instability in $L^2$
(the energy space, large spatial scale) were based on the vorticity
formulation and therefore only work in 2-dim. In this talk, we prove,
in any dimensions, the existence of the unique local unstable manifold
of $v_*$, under certain conditions, and thus its nonlinear
instability. Our approach is based on the observation that the Euler
equation on a fixed domain is an ODE on an infinite dimensional
manifold of volume preserving maps in function spaces. This is a
joint work with Zhiwu Lin.
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.