Local Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces

By Zihua Guo. 2008

58 citations recorded by the source corpus.

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation \[\partial_t u+|\partial_x|^{1+\alpha}\partial_x u+uu_x=0,\ u(x,0)=u_0(x),\] is locally well-posed in the Sobolev spaces $H^s$ for $s>1-\alpha$ if $0\leq \alpha \leq 1$. The new ingredient is that we develop the methods of Ionescu, Kenig and Tataru \cite{IKT} to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and Tzvetkovin \cite{MST}. Moreover, as a bi-product we prove that if $0<\alpha \leq 1$ the corresponding modified equation (with the nonlinearity $\pm uuu_x$) is locally well-posed in $H^s$ for $s\geq 1/2-\alpha/4$.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/50f3acfdcc98c4c47192ac881f0f1e34183c3511

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Improvement of the energy method for strongly non resonant dispersive equations and applications (2014, unknown): “As far as we know, the best available results without gauge transformation are obtained in [Guo 2012], where the LWP in H .”
The IVP for the dispersion generalized Benjamin–Ono equation in weighted Sobolev spaces (2011, background): “olev spaces Hs(R) for (1.1) cannot be obtained by an argument based only on the contraction principle. Local well-posedness in classical Sobolev spaces for (1.1) has been studied in [29], [19], [37], [18], and [20] where local well-posedness was established for s≥ 0. Real solutions of the IVP (1.1) ”
Well-posedness for the fifth-order KdV equation in the energy space (2012, methodology): “We refer to [4], [23] for previous works using similar spaces to prove a priori bounds for the 1D cubic NLS at low regularity and also to [11], [12], [29] for applications to other dispersive equations.”