Probabilistic well-posedness for the cubic wave equation

By Nicolas Burq, Nikolay Tzvetkov. 2011

155 citations recorded by the source corpus.

The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs measures), and second, we show that the flow that we are able to construct enjoys very nice dynamical properties, including a new notion of probabilistic continuity.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b234c5f25222b77fb03446f68f847982c1946d1d

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In Statistical and Nonlinear Physics

Citations in the corpus

Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity (2018, result): “This is analogous to the situation for the recent study of nonlinear dispersive PDEs with randomized initial data [14, 15, 60, 7, 80, 73, 74, 8], where a set of probability one for local-in-time or global-in-time well-posedness depends on the choice of deterministic initial data (to which a randomiz”
Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^d$, $d=4$ and $5$ (2014, unknown): “y of the nonlinear part vω of the solution uωon each ﬁnite time interval. See Proposition 5.2. The probabilistic energy bound that we use here is the analogue of that obtained by Burq and Tzvetkov in [19] for the cubic NLW on T3. In [6], the author with Bényi and Oh applied a similar probabilistic p”
On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3 (2014, unknown): “We point out that the upper bound R(T, ε) in [15] tends to ∞ as T → ∞.”
Random Data Cauchy Theory for Nonlinear Wave Equations of Power-Type on ℝ3 (2014, methodology): “2 of [12] to prove that if 0 < s < 1 and f = f1 f2 ∈ H x 3 ×Hs−1 x 3 , then for almost every ∈ there exists a unique global solution”
Almost sure well-posedness for the periodic 3D quintic nonlinear Schr\"odinger equation below the energy space (2013, background): “[23] Herr, S., Tataru, D. and Tzvetkov, N. Global well-posedness of the energy-critical nonlinear Schrdinger equation with small initial data in H1(T3).”
WIENER RANDOMIZATION ON UNBOUNDED DOMAINS AND AN APPLICATION TO ALMOST SURE WELL-POSEDNESS OF NLS (2014, background): “In the context of the defocusing cubic NLW on R4, one can obtain an a priori bound on the energy of the nonlinear part of a solution, see [16].”