We prove the local well-posedness for the Cauchy problem of Korteweg-de Vries equation in a quasi periodic function space. The function space contains functions satisfying f=f1+f2+...+fN where fj is in the Sobolev space of order s > −1/2N of aj periodic functions. Note that f is not periodic when the ratio of periods ai/aj is irrational. We also prove an ill-posedness result in the sense that the flow map (if it exists) is not C2, which is related to the Diophantine problem. We also prove the global well-posedness in an almost periodic function space.
In this talk, we consider the uniqueness of stationary solutions to the Navier-Stokes equation in 3-dimensional exterior domains within the class u ∈ L3,∞ with ∇u ∈ L3/2,∞, where L3,∞ and L3/2,∞ are the Lorentz spaces. We show that if u and v are solutions, and if u ∈ Lp for some p > 3 and v is sufficiently small in L3,∞, then u=v. The proof is based on the regularity theory of the perturbed Stokes equations and the bootstrap argument.
We will talk about the time global wellposedness in Lp for the Chern-Simons-Dirac equation in 1 dimension. We apply the standard iteration arguments to obtain the time local solution. We derive some a priori estimates to extend it to the global time. For the critical case in L1, we need to deny the possibility of the mass concentration phenomena of the solutions. This is a joint work with Takayoshi Ogawa (Tohoku university).
In multi-connected domains, it is still an open question whether there does exist a solution of the stationary Navier-Stoeks equations with the inhomogeneous boundary data whose total flux is zero. The relation between the nonlinear structure of the equations and the topological invariance of the domain plays an important role for the solvability of this problem. We prove that if the harmonic part of solenoidal extensions of the given boundary data associated with the second Betti number of the domain is orthogonal to non-trivial solutions of the Euler equations, then there exists a solution for any viscosity constant. The relation between Leary's inequality and the topological type of the domain is also clarified. This talk is based on the joint work with Prof. Taku Yanagisawa at Nara Women University.
We consider the global existence of L2 solutions for the 1D Zakharov equations with additive noise. The 1D Zakharov equations with additive noise are proposed to model the Langmuir turbulence in the ionosphere. We employ the argument by Colliander, Holmer and Tzirakis to prove the global existence of solutions for the Cauchy problem with Schrödinger part in L2 and wave part in Ḣ −1/2.
Considering viscous (scalar) conservation laws, there is a strong connection between decay estimates and uniqueness of the zero-viscosity limit (to the hyperbolic nonlinear solution). In fact, in the case of convex flux, the Oleinik entropy condition is actually a decay estimate. The talk will consist of two parts:
(i) Decay estimates in the multi-dimensional case (even with boundaries).
(ii) Decay estimates in the one-dimensional case, where some "classical" estimates are revisited and open problems remain in the nonconvex case.