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A continuum of invariant measures for the periodic KdV and mKdV equations
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abstract
We consider the real-valued defocusing modified Korteweg-de Vries equation (mKdV) on the circle. Based on the complete integrability of mKdV, Killip-Vi\c{s}an-Zhang (2018) discovered a conserved quantity which they used to prove low regularity a priori bounds for solutions. It has been an open question if this conserved quantity can be used to define invariant measures supported at fractional Sobolev regularities. Motivated by this question, we construct probability measures supported on $H^s(\mathbb{T})$ for $0<s<1/2$ invariant under the mKdV flow. We then use the Miura transform to obtain invariant measures for the Korteweg-de Vries equation, whose supports are rougher than the white noise measure. We also obtain analogous results for the defocusing cubic nonlinear Schr\"{o}dinger equation. These invariant measures cover the lowest possible regularities for which the flows of these equations are well-posed.
Forward citations
Cited by 2 Pith papers
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Well-posedness and invariant measures for complex valued modified KdV equation
For the complex-valued periodic mKdV, the authors construct infinitely many invariant weighted Gaussian measures and prove unconditional well-posedness at H^s for s>4/3.
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New invariant surface measures for the cubic Schr\"odinger equation
The authors construct invariant probability measures supported on level sets of the renormalized mass for the defocusing cubic nonlinear Schrodinger equation on the one- and two-dimensional torus.
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