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Register a new userGlobal well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation
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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.
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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.
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, introduced 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.
In this paper, KdV-type equations with time- and space-dependent coefficients are considered. Assuming that the dispersion coefficient in front of $u_{xxx}$ is positive and uniformly bounded away from the origin and that a primitive function of the ratio 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.
In this paper we consider two numerical scheme based on trapezoidal rule in time for the linearized KdV equation in one space dimension. The goal is to derive some suitable artificial boundary conditions for these two full discretization using Z-transformation. We give some numerical benchmark examples from the literature to illustrate our findings.
Considering the Cauchy problem for the modified finite-depth-fluid equation $partial_tu-G_delta(partial_x^2u)mp u^2u_x=0, u(0)=u_0$, where $G_delta f=-i ft ^{-1}[coth(2pi delta xi)-frac{1}{2pi delta xi}]ft f$, $deltages 1$, and $u$ is a real-valued function, we show that it is uniformly globally well-posed if $u_0 in H^s (sgeq 1/2)$ with $ orm{u_0}_{L^2}$ sufficiently small for all $delta ges 1$. Our result is sharp in the sense that the solution map fails to be $C^3$ in $H^s (s<1/2)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the modified Benjamin-Ono equation if $delta$ tends to $+infty$.
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