On universality of critical behaviour in Hamiltonian PDEs

临界对比偏微分方程的有效行为：微共振、频率转换与时间色散特性。II

Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. II

本研究针对完整线性弹性情形下的双孔隙度模型，建立阶数精确理论，并揭示高振荡弹性复合材料的时间与频率色散特性。

Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. I

A novel approach to critical-contrast homogenisation for periodic PDEs is proposed, via an explicit asymptotic analysis of Dirichlet-to-Neumann operators. Norm-resolvent asymptotics for non-uniformly elliptic problems with highly oscillating coefficients are explicitly constructed. An essential feature of the new technique is that it relates homogenisation limits to a class of time-dispersive media.

On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the {\it tritronquée} solution to the Painlevé-I equation

We argue that the critical behaviour near the point of ``gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation $ iεψ_t +\frac{ε^2}2ψ_{xx}+ |ψ|^2 ψ=0$ with analytic initial data of the form $ψ(x,0;ε) =A(x) e^{\frac{i}ε S(x)}$ is approximately described by a particular solution to the Painlevé-I equation.

Variational approach to complicated similarity solutions pf higher-order nonlinear PDEs. I

Three classes of higher-order nonlinear parabolic hyperbolic, and nonlinear dispersion equations are shown to admit exact blow-up or compacton solutions, which are induced by elliptic equations with non-Lipschitz nonlinearities. Variational techniques give countable families of various compactly supported solutions with oscllatory behaviour cloase to the interfaces. Overall, the whole set of such compact patterns reveals certain chaotic properties and contains solutions of arbitrary complexity.

Birkhoff Normal Form and Hamiltonian PDEs

These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some perturbations technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. We are in particular interested with stability results. Our approach is centered on the Birkhoff normal form theorem that we first proved in finite dimension. Then, after giving some exemples of Hamiltonian PDEs, we present an abstract Birkhoff normal form theorem in infinite dimen…

Boundary regularity of stochastic PDEs

The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly - and in a sense, arbitrarily - bad: as shown by Krylov, for any $α>0$ one can find a simple $1$-dimensional constant coefficient linear equation whose solution at the boundary is not $α$-Hölder continuous. We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on $C^1$ domains are proved to be $α$-Hölder continuous up to the boundary with some $α>0$.