On the Probabilistic Cauchy Theory for Nonlinear Dispersive PDEs

Abstract

In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.2010 Mathematics Subject Classification. 35Q55, 35L71, 60H30.

Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

“…The literature on random dispersive partial differential equations is vast. We refer the interested reader to the survey [6], and mention the related works [2,4,8,9,11,12,14,15,16,38,39,41,42,44]. In the following discussion, we focus on the Wiener randomization [3,38] of a function f P H s pR d q.…”

“…Using the deterministic well-posedness theorem and stability theory, it can be shown (cf. [26,44]) that the solution to (6) exists as long as the energy of v remains bounded. Of course, due to the forcing term in (6), the energy is no longer conserved.…”

“…[26,44]) that the solution to (6) exists as long as the energy of v remains bounded. Of course, due to the forcing term in (6), the energy is no longer conserved. In addition, a global bound on the energy of v implies a global bound on the L 3 t L 6…”

Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion

“…On the other hand, in the field of Hamiltonian PDEs, Gaussian measures naturally appear in the construction of invariant measures associated to conservation laws such as Gibbs measures, starting with the seminal work of Bourgain [7,8]. See [29,4] for the references therein. In [36], the third author initiated the study of transport properties of Gaussian measures under the flow of a Hamiltonian PDE, where two methods were presented in establishing quasi-invariance of the Gaussian measures µ s as stated in Theorem 1.2.…”

“…See Lemma 2.5. We also note that the phase function ψ(n) in (1. Given t, τ ∈ R, let Φ(t) : L 2 → L 2 be the solution map for (1.1) and Ψ 0 (t, τ ) and Ψ 1 (t, τ ) : L 2 → L 2 be the solution maps for (1.11) and (1.18), respectively, sending initial data at time τ to solutions at time t. 4 When τ = 0, we denote Ψ 0 (t, 0) and Ψ 1 (t, 0) by Ψ 0 (t) and Ψ 1 (t) for simplicity. Then, it follows from (1.7), (1.10), (1.14), and (1.17) that…”