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تسجيل مستخدم جديدGlobal Well-posedness of Korteweg-de Vries equation in $H^{-3/4}(R)$
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Zihua Guo
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We prove that the Korteweg-de Vries initial-value problem is globally well-posed in $H^{-3/4}(R)$ and the modified Korteweg-de Vries initial-value problem is globally well-posed in $H^{1/4}(R)$. The new ingredient is that we use directly the contraction principle to prove local well-posedness for KdV equation at $s=-3/4$ by constructing some special resolution spaces in order to avoid some logarithmic divergence from the high-high interactions. Our local solution has almost the same properties as those for $H^s (s>-3/4)$ solution which enable us to apply the I-method to extend it to a global solution.
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اقرأ أيضاً
Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation begin{eqnarray*} u_t+u_{xxx}+epsilon |partial_x|^{2alpha}u+(u^2)_x=0, u(0)=phi, end{eqnarray*} where $0<epsilon,alphaleq 1$ and $u$ is a real-valued function, we show that it
is globally well-posed in $H^s (s>s_alpha)$, and uniformly globally well-posed in $H^s (s>-3/4)$ for all $epsilon in (0,1)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the KdV equation if $epsilon$ tends to 0.
We prove that the Cauchy problem for the Schrodinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sovolev spaces $L^2(R)times H^{-{3/4}}(R)$. The new ingredient is that we use the $bar{F}^s$ type space, introdu
ced by the first author in cite{G}, to deal with the KdV part of the system and the coupling terms. In order to overcome the difficulty caused by the lack of scaling invariance, we prove uniform estimates for the multiplier. This result improves the previous one by Corcho and Linares.
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atio between the anti-dissipation and the dispersion coefficients is bounded from below, we prove the existence and uniqueness of a solution $u$ such that $h u$ belongs to a classical Sobolev space, where $h$ is a function related to this ratio. The LWP in $H^s(mathbb{R})$, $s>1/2$, in the classical (Hadamard) sense is also proven under an assumption on the integrability of this ratio. Our approach combines a change of unknown with dispersive estimates. Note that previous results were restricted to $H^s(mathbb{R})$, $s>3/2$, and only used the dispersion to compensate the anti-dissipation and not to lower the Sobolev index required for well-posedness.
We prove global well-posedness of the fifth-order Korteweg-de Vries equation on the real line for initial data in $H^{-1}(mathbb{R})$. By comparison, the optimal regularity for well-posedness on the torus is known to be $L^2(mathbb{R}/mathbb{Z})$.
In order to prove this result, we develop a strategy for integrating the local smoothing effect into the method of commuting flows introduced previously in the context of KdV. It is this synthesis that allows us to go beyond the known threshold on the torus.
In this remark, we give another approach to the local well-posedness of quadratic Schrodinger equation with nonlinearity $ubar u$ in $H^{-1/4}$, which was already proved by Kishimoto cite{kis}. Our resolution space is $l^1$-analogue of $X^{s,b}$ spac
e with low frequency part in a weaker space $L^{infty}_{t}L^2_x$. Such type spaces was developed by Guo. cite{G} to deal the KdV endpoint $H^{-3/4}$ regularity.
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