On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916128818724864 |
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| author | Stewart, Gavin |
| author_facet | Stewart, Gavin |
| contents | We consider a periodic nonlinear Schrödinger equation with white noise dispersion and a power nonlinearity given by
\begin{equation*}
idu = Δu \circ dW_t + |u|^{p-1}u\;dt
\end{equation*}
By proving stochastic Strichartz estimates, we are able to prove almost sure global wellposedness of this equation with $L^2$ initial data for nonlinearities with exponent $1 < p \leq 3$. By generalizing the Fourier restriction spaces $X^{s,b}$ to the stochastic setting, we also prove that our solutions agree with the ones constructed by Chouk and Gubinelli using rough path techniques. We also consider the quintic equation ($p=5$), and show that it is analytically illposed in $L^1_ωC_t L^2_x$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_03391 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion Stewart, Gavin Analysis of PDEs Probability 35R60, 35Q55 We consider a periodic nonlinear Schrödinger equation with white noise dispersion and a power nonlinearity given by \begin{equation*} idu = Δu \circ dW_t + |u|^{p-1}u\;dt \end{equation*} By proving stochastic Strichartz estimates, we are able to prove almost sure global wellposedness of this equation with $L^2$ initial data for nonlinearities with exponent $1 < p \leq 3$. By generalizing the Fourier restriction spaces $X^{s,b}$ to the stochastic setting, we also prove that our solutions agree with the ones constructed by Chouk and Gubinelli using rough path techniques. We also consider the quintic equation ($p=5$), and show that it is analytically illposed in $L^1_ωC_t L^2_x$. |
| title | On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion |
| topic | Analysis of PDEs Probability 35R60, 35Q55 |
| url | https://arxiv.org/abs/2208.03391 |