On the regularization mechanism for the periodic Korteweg–de Vries equation

By A. Babin, Alexei A. Ilyin, Edriss S. Titi. 2009

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In this paper we develop and use successive averaging methods for explaining the regularization mechanism in the the periodic Korteweg–de Vries (KdV) equation in the homogeneous Sobolev spaces Ḣs for s ≥ 0. Specifically, we prove the global existence, uniqueness, and Lipschitz‐continuous dependence on the initial data of the solutions of the periodic KdV. For the case where the initial data is in L2 we also show the Lipschitz‐continuous dependence of these solutions with respect to the initial data as maps from Ḣs to Ḣs for s ∈(−1,0]. © 2010 Wiley Periodicals, Inc.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5f9319afd4aa3f6cedc4af32e2f7401dde857e43

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In Control and Systems Engineering

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schr\"odinger equation (2017, methodology): “y the Hs-energy functional kuk2 Hs. In the following, we prove Theorem 1.2 by performing normal form reductions inﬁnitely many times. Our normal form approach is analogous to the approach employed in [1, 27, 18]. In particular, in [18], the ﬁrst author (with Guo and Kwon) implemented an inﬁnite iter”
Smoothing and global attractors for the Zakharov system on the torus (2012, methodology): “The result will follow from a combination of the method of normal forms (through differentiation by parts) inspired by the result in [4], and the restricted norm method of Bourgain, [7].”
Averaging along irregular curves and regularisation of ODEs (2012, methodology): “or any continuous path x ∈ Qγ W and is independent of the speciﬁc sequence (bn)n≥1. ii) almost surely for all bin FLα+1 there exists a solution x∈ Qγ W of the equation xt = x0 + Zt 0 b(s,xs)ds+wt, t∈ [0,1] where the integral of bis understood according the limiting procedure given in eq. (2) above. ”
Improvement of the energy method for strongly non resonant dispersive equations and applications (2014, unknown): “For Lα+1 := ∂ 3 x, we recover the unconditional LWP result in L2(T) obtained in [2] for the KdV equation.”
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