Invariant measure for a three dimensional nonlinear wave equation

By Nicolas Burq, Nikolay Tzvetkov. 2007

66 citations recorded by the source corpus.

We study the long time behavior of the subcritical (subcubic) defocussing nonlinear wave equation on the three dimensional ball, for random data of low regularity. We prove that for a large set of radial initial data in $\cap_{s<1/2} H^s(B(0,1))$ the equation is (globally in time) well posed and we construct an invariant measure.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/235a75eb646919f3f1d419835ecd7353a7e60c52

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Almost sure well-posedness of the cubic nonlinear Schr\ (2009, background): “) See Bourgain [4, 5], Burq-Tzvetkov [9, 11], Oh [24, 25, 27], and Tzvetkov [33, 34].”
Invariant Gibbs measures and a.s. global well posedness for coupled KdV systems (2009, background): “Recently, Burq-Tzvetkov [6] independently and simultaneously used similar ideas to prove a.s. GWP for the nonlinear wave equation on the unit ball in R3 under the radial symmetry.”
Random data Cauchy theory for supercritical wave equations II: a global existence result (2007, methodology): “Introduction In our previous work [6], we developed a general theory for constructing local strong solutions to nonlinear wave equations, posed on compact riemannian manifolds with supercritical random initial data.”
Invariance of the White Noise for KdV (2009, background): “Also, see [6], [19], [23], [24] for the details.”
Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^d$, $d=4$ and $5$ (2014, unknown): “orous by exploiting the invariance of the ﬁnite dimensional Gibbs measures for the associated ﬁnite dimensional approximations of NLS. This approach has been used in subsequent works; see for example [8, 9, 65, 16, 18, 50, 45, 26, 14, 24, 57, 12]. In our context, however, there is no apparent (forma”
Invariance of the Gibbs measure for the periodic quartic gKdV (2012, background): “For other works that have used nonlinear smoothing to establish local dynamics in the support of measures on phase space, see (for example) BurqTzvetkov [5, 6, 7], Oh [26], Colliander-Oh [10], and Nahmod-Pavlović-Staffilani [21].”
INVARIANT WEIGHTED WIENER MEASURES AND ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE PERIODIC DERIVATIVE NLS. (2010, methodology): “.S. GWP FOR DNLS 9 The following lemma gives control on how close the ﬁnite dimensional approximations are to the solution of (2.8). Our proof is a variation of Bourgain’s Lemma 2.27 in [3] (see also [12]). Lemma 3.3 (Approximation lemma). Let v0 ∈ FLs,r(T),s>1 2,2 <r<4 be such that kv0kFLs”