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Register a new userOn well-posedness for some Korteweg-De Vries type equations with variable coefficients
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Luc Molinet
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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.
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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.
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 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.
In this paper we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg-de Vries-type (KdV-type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.
In this paper we study the asymptotics of the Korteweg--de Vries (KdV) equation with steplike initial data, which leads to shock waves, in the middle region between the dispersive tail and the soliton region, as $t rightarrow infty$. In our previous work we have dealt with this question, but failed to obtain uniform estimates in $x$ and $t$ because of the previously unknown singular behaviour of the matrix model solution. The main goal of this paper is to close this gap. We present an alternative approach to the usual argument involving a small norm Riemann--Hilbert (R-H) problem, which is based instead on Fredholm index theory for singular integral operators. In particular, we avoid the construction of a global model matrix solution, which would be singular for arbitrary large $x$ and $t$, and utilize only the symmetric model vector solution, which always exists and is unique.
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