An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation. (Régularité optimale pour la quasi-invariance de mesures gaussiennes par le flot de l’équation de Schrödinger non linéaire d’ordre 4.) (English. French summary) Zbl 1405.35198
Summary: We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we establish an optimal regularity result for quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces. The main new ingredient is an improved energy estimate established by performing an infinite iteration of normal form reductions on the energy functional. Furthermore, we show that the dispersion is essential for such a quasi-invariance result by proving non quasi-invariance of the Gaussian measures under the dynamics of the dispersionless model.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35R06 | PDEs with measure |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Keywords:
fourth order nonlinear Schrödinger equation; biharmonic nonlinear Schrödinger equation; quasi-invariant measure; normal form methodReferences:
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