Global well-posedness of Korteweg-de Vries equation in H^-3/4 (R)

By Zihua Guo. 2008

184 citations recorded by the source corpus.

Abstract 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 in H − 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.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/24bc494f5554b215c964a0b1e18ed83f4114267f

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Lower Bounds on the Radius of Spatial Analyticity for the KdV Equation (2015, unknown): “By the I-method, it has been shown that global well-posedness in fact holds for all s ≥ −3/4 (see [5,8]).”
Local Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces (2008, background): “Recently, the author [ 7 ] obtained global well-posedness in H −3/4 by”
KdV is well-posed in H–1 (2018, background): “solution map to be uniformly continuous on bounded sets; see [10]. These local well-posedness results were made global in time in [12], excepting the endpoint case H−3/4(R), which was proved later in [22, 38]. At that time, no exact conservation laws were known that were adapted to negative regulari”
A note on ill posedness for the KdV equation (2010, background): “[10] or [4]) and su(mKdV ) = 1/4 for the mKdV equations (cf.”
Long time behavior of solutions to the mKdV (2014, background): “e Cauchy problem for (1.1) has been studied extensively. For a summary of known results we refer the reader to [29]. In particular, the mKdV is both locally well-posed [23–25] and globally well-posed [3,12,27] in Hs for s≥ 1 4. Below s= 1 4 the solution map fails to be uniformly continuous [1,26] al”