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Register a new userLocal Well-posedness and a priori bounds for the modified Benjamin-Ono equation without using a gauge transformation
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Zihua Guo
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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 $s>1/4$.
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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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The Benjamin Ono equation with a slowly varying potential is $$ text{(pBO)} qquad u_t + (Hu_x-Vu + tfrac12 u^2)_x=0 $$ with $V(x)=W(hx)$, $0< h ll 1$, and $Win C_c^infty(mathbb{R})$, and $H$ denotes the Hilbert transform. The soliton profile is $$Q_{a,c}(x) = cQ(c(x-a)) ,, text{ where } Q(x) = frac{4}{1+x^2}$$ and $ain mathbb{R}$, $c>0$ are parameters. For initial condition $u_0(x)$ to (pBO) close to $Q_{0,1}(x)$, it was shown in a previous work by Z. Zhang that the solution $u(x,t)$ to (pBO) remains close to $Q_{a(t),c(t)}(x)$ and approximate parameter dynamics for $(a,c)$ were provided, on a dynamically relevant time scale. In this paper, we prove exact $(a,c)$ parameter dynamics. This is achieved using the basic framework of the previous work by Z. Zhang but adding a local virial estimate for the linearization of (pBO) around the soliton. This is a local-in-space estimate averaged in time, often called a local smoothing estimate, showing that effectively the remainder function in the perturbation analysis is smaller near the soliton than globally in space. A weaker version of this estimate is proved in a paper by Kenig & Martel as part of a ``linear Liouville result, and we have adapted and extended their proof for our application.
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