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Probabilistic well-posedeness for the nonlinear Schr\"odinger equation on the 2d sphere I: positive regularities
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Probabilistic well-posedeness for the nonlinear Schr\"odinger equation on the $2d$ sphere I: positive regularities
abstract
We establish the probabilistic well-posedness of the nonlinear Schr\"odinger equation on the $2d$ sphere $\mathbb{S}^{2}$. The initial data are distributed according to Gaussian measures with typical regularity $H^{s}(\mathbb{S}^{2})$, for $s>0$. This level of regularity goes significantly beyond existing deterministic results, in a regime where the flow map cannot be extended uniformly continuously.
Forward citations
Cited by 4 Pith papers
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Gauge transforms, random averaging operator ansatz and improved probabilistic well-posedness for the radial NLS on the $3d$ ball
Constructs probabilistic strong solutions to radial cubic NLS on 3D ball in supercritical regime via gauge transforms and random averaging operators, improving Bourgain-Bulut.
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On the pointwise convergence of NLS flow on $ \S^2 $
The cubic NLS on S² converges pointwise almost everywhere to initial data almost surely at low regularity, and a new necessary condition is given for L^p maximal estimates of the linear Schrödinger equation on S².
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Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation
Renormalized BBM with rough Gaussian initial data converges in law to stochastic BBM forced by derivative of spatial white noise, for all regularities alpha <= 1/4.
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On probabilistic ill-posedness
The paper defines enhanced probabilistic well-posedness by imposing stability at the origin and reinterprets recent 'beyond variance blowup' results for dispersive PDEs as probabilistic ill-posedness.
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