Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.18184 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929607266009088 |
|---|---|
| author | Casteras, Jean-baptiste Földes, Juraj Oliveira, Itamar Uraltsev, Gennady |
| author_facet | Casteras, Jean-baptiste Földes, Juraj Oliveira, Itamar Uraltsev, Gennady |
| contents | In this paper, we study the local well-posedness of the cubic Schrödinger equation $$(i\partial_t + \mathcal{L}) u = \pm |u|^2 u \qquad \textrm{on} \quad \ I\times \mathbb{R}^d ,$$ with initial data being a Wiener randomization at unit scale of a given function $f$ and $\mathcal{L}$ being an operator of degree $σ\geq 2$. In particular, we prove that a solution exists almost-surely locally in time provided $f\in H^{S}_{x}(\mathbb{R}^{d})$ with $S>\frac{2-σ}{4}$ for $d\leq \frac{3σ}{2}$, i.e. even if the initial datum is taken in certain negative order Sobolev spaces. The solutions are constructed as a sum of an explicit multilinear expansion of the flow in terms of the random initial data and of an additional smoother remainder term with deterministically subcritical regularity. We develop the framework of directional space-time norms to control the (probabilistic) multilinear expansion and the (deterministic) remainder term and to obtain improved bilinear probabilistic-deterministic Strichartz estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18184 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Probabilistic well-posedness of generalized cubic nonlinear Schrödinger equations with strong dispersion using higher order expansions Casteras, Jean-baptiste Földes, Juraj Oliveira, Itamar Uraltsev, Gennady Analysis of PDEs Probability In this paper, we study the local well-posedness of the cubic Schrödinger equation $$(i\partial_t + \mathcal{L}) u = \pm |u|^2 u \qquad \textrm{on} \quad \ I\times \mathbb{R}^d ,$$ with initial data being a Wiener randomization at unit scale of a given function $f$ and $\mathcal{L}$ being an operator of degree $σ\geq 2$. In particular, we prove that a solution exists almost-surely locally in time provided $f\in H^{S}_{x}(\mathbb{R}^{d})$ with $S>\frac{2-σ}{4}$ for $d\leq \frac{3σ}{2}$, i.e. even if the initial datum is taken in certain negative order Sobolev spaces. The solutions are constructed as a sum of an explicit multilinear expansion of the flow in terms of the random initial data and of an additional smoother remainder term with deterministically subcritical regularity. We develop the framework of directional space-time norms to control the (probabilistic) multilinear expansion and the (deterministic) remainder term and to obtain improved bilinear probabilistic-deterministic Strichartz estimates. |
| title | Probabilistic well-posedness of generalized cubic nonlinear Schrödinger equations with strong dispersion using higher order expansions |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2411.18184 |