Random data Cauchy theory for supercritical wave equations II: a global existence result

By Nicolas Burq, Nikolay Tzvetkov. 2007

310 citations recorded by the source corpus.

We prove that the subquartic wave equation on the three dimensional ball Θ, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in $\bigcap_{s<1/2} (H^s(\Theta)\times H^{s-1}(\Theta))$. We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work [6] and invariant measure considerations, inspired by earlier works by Bourgain [2, 3] on the non linear Schrödinger equation, which allow us to obtain also precise large time dynamical informations on our solutions.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/46abe0815ae5a73cf9e2e0be6152e104fa1b530d

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 Geometry and Topology

In Mathematical Physics

Citations in the corpus

Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on $\mathbb{R}^3$ (2015, background): “Following the works of Bourgain [6] and Burq-Tzvetkov [9], there have been many results on probabilistic construction of solutions to evolution equations via randomization of initial data.”
Injections de Sobolev probabilistes et applications (2011, background): “– Des résultats similaires apparaissent dans un cadre légèrement différent dans [12].”
Random Data Cauchy Theory for Nonlinear Wave Equations of Power-Type on ℝ3 (2014, methodology): “This follows by a similar argument to the one found in Appendix B of [10], using (1.”
Optimal Shape and Location of Sensors for Parabolic Equations with Random Initial Data (2014, methodology): “For any given y0 ∈ D(A0), the Fourier coefficients of y0, defined by (13), are randomized by defining aνj = βνj a j for every j ∈ N∗, where (βνj ) j∈N∗ is a sequence of independent real random variables on a probability space (X ,F ,P) having mean equal to 0, variance equal to 1, and a super exponen”
Global Dynamics for the Two-dimensional Stochastic Nonlinear Wave Equations (2020, methodology): “In this case, we can apply the globalization argument by Burq-Tzvetkov [5], originally introduced in the context of the cubic NLW on T3 with random initial data.”
Probabilistic well-posedness for the cubic wave equation (2011, unknown): “with a form of continuity of the flow, as opposed to the mere global existence results as developed in our previous works [11, 9] or in [5, 14] (notice however that the existence of Gibbs measures [5, 11, 9] in itself is an interesting dynamical property of the flow).”
WIENER RANDOMIZATION ON UNBOUNDED DOMAINS AND AN APPLICATION TO ALMOST SURE WELL-POSEDNESS OF NLS (2014, background): “Burq-Tzvetkov [15] further explored the study of Cauchy problems with more general random initial data.”
On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3 (2014, background): “See Burq-Tzvetkov [15].”
Almost sure well-posedness for the periodic 3D quintic nonlinear Schr\"odinger equation below the energy space (2013, unknown): “1), then w = u−S(t)φω is smoother; see also [11, 17, 25].”
Invariance of the Gibbs measure for the Benjamin-Ono equation (2012, background): “We may then prove the same bound (possibly with some OC1(1) factors) for P>K(u (5) − u([5])), P≤K(u((5)) − u([5])), P≤K(u((4)) − u([4])), P>K(u((4)) − u([4])) and v((4)) − v([4]) in that order, in the same way as in Section 11 (note T−1 is assumed to be larger than any power of K).”
Remarks on the Gibbs measures for nonlinear dispersive equations (2014, background): “See [4, 5, 54, 49, 50, 17, 39, 40, 13] where this approach has been fruitful.”
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): “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”