The nonlinear Schrödinger equation in the weakly nonlinear regime with random Gaussian fields as initial data is considered. The problem is set on the torus in any dimension greater than two. A conjecture in statistical physics is that there exists a kinetic time scale depending on the frequency localisation of the data and on the strength of the nonlinearity, on which the expectation of the squares of moduli of Fourier modes evolve according to an effective equation: the so-called kinetic wave equation. When the kinetic time for our setup is 1, we prove this conjecture up to an arbitrarily small polynomial loss. When the kinetic time is larger than 1, we obtain its validity on a more restricted time scale. The key idea of the proof is the use of Feynman interaction diagrams both in the construction of an approximate solution and in the study of its nonlinear stability. We perform a truncated series expansion in the initial data, and obtain bounds in average in various function spaces for its elements. The linearised dynamics then involves a linear Schrödinger equation with a corresponding random potential. We bound the expectation of the operator norm in Bourgain spaces using diagrams and random matrix tools. This gives a new approach for the analysis of nonlinear wave equations out of equilibrium, and gives hope that refinements of the method could help settle the conjecture.

We consider Burgers equation with transverse viscosityWe construct and describe precisely a family of solutions which become singular in finite time by having their gradient becoming unbounded. To leading order, the solution is given by a backward self-similar solution of Burgers equation along the x variable, whose scaling parameters evolve according to parabolic equations along the y variable, one of them being the quadratic semi-linear heat equation. We develop a new framework adapted to this mixed hyperbolic/parabolic blow-up problem, revisit the construction of flat blow-up profiles for the semi-linear heat equation, and the self-similarity in the shocks of Burgers equation.

No abstract

Abstract. We consider the semilinear heat equation in large dimension d ≥ 11 ∂tu = ∆u + |u| p−1 u, p = 2q + 1, q ∈ N on a smooth bounded domain Ω ⊂ R d with Dirichlet boundary condition. In the supercritical rangewe prove the existence of a countable family (u ℓ ) ℓ∈N of solutions blowing-up at time T > 0 with type II blow up:. They concentrate the ground state Q being the only radially and decaying solution of ∆Q + Q p = 0:at some point x0 ∈ Ω. The result generalizes previous works on the existence of type II blow-up solutions, either constructive [14,15,35] or nonconstructive [25,36], which only existed in the radial setting and relied on parabolic arguments. The present proof uses robust nonlinear tools instead, based on energy methods and modulation techniques in the continuity of [3,32]. This is the first non-radial construction of a solution blowing up by concentration of a stationary state in the supercritical regime, and provides a general strategy to prove similar results for dispersive equations or parabolic systems and to extend it to multiple blow ups.

We consider the energy super critical semilinear heat equationWe first revisit the construction of radially symmetric self similar solutions performed through an ode approach in [51], [2], and propose a bifurcation type argument suggested in [3] which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. We then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional non radial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self similar blow up in non radial energy super critical settings. (T − t)1 p−1 2010 Mathematics Subject Classification. primary, 35K58 35B44 35B35, secondary 35J61 35B32.