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Register a new userLocal Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces
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Authors
Zihua Guo
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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 $0leq alpha leq 1$. 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 $sgeq 1/2-alpha/4$.
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In this work we continue our study initiated in cite{GFGP} on the uniqueness properties of real solutions to the IVP associated to the Benjamin-Ono (BO) equation. In particular, we shall show that the uniqueness results established in cite{GFGP} do not extend to any pair of non-vanishing solutions of the BO equation. Also, we shall prove that the uniqueness result established in cite{GFGP} under a hypothesis involving information of the solution at three different times can not be relaxed to two different times.
We prove that the complex-valued modified Benjamin-Ono (mBO) equation is locally wellposed if the initial data $phi$ belongs to $H^s$ for $sgeq 1/2$ with $ orm{phi}_{L^2}$ sufficiently small without performing a gauge transformation. Hence the real-valued mBO equation is globally wellposed for those initial datas, which is contained in the results of C. Kenig and H. Takaoka cite{KenigT} where the smallness condition is not needed. We also prove that the real-valued $H^infty$ solutions to mBO equation satisfy a priori local in time $H^s$ bounds in terms of the $H^s$ size of the initial data for $s>1/4$.
We prove the discontinuity for the weak $ L^2(T) $-topology of the flow-map associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in $ H^s(T) $ as soon as $ s<0 $ and thus completes exactly the well-posedness result obtained by the author.
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We study the Cauchy problem in $n$-dimensional space for the system of Navier-Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical mixed-norm Lebesgue spaces. Being in the mixed-norm Lebesgue spaces, both of the initial data and the solutions could be singular at certain points or decaying to zero at infinity with different rates in different spatial variable directions. Some of these singular rates could be very strong and some of the decaying rates could be significantly slow. Besides other interests, the results of the paper particularly show an interesting phenomena on the persistence of the anisotropic behavior of the initial data under the evolution. To achieve the goals, fundamental analysis theory such as Youngs inequality, time decaying of solutions for heat equations, the boundedness of the Helmholtz-Leray projection, and the boundedness of the Riesz tranfroms are developed in mixed-norm Lebesgue spaces. These fundamental analysis results are independently topics of great interests and they are potentially useful in other problems.
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