On the Cauchy problem for higher-order nonlinear dispersive equations

By Didier Pilod. 2007

47 citations recorded by the source corpus.

Abstract We study the higher-order nonlinear dispersive equation ∂ t u + ∂ x 2 j + 1 u = ∑ 0 ⩽ j 1 + j 2 ⩽ 2 j a j 1 , j 2 ∂ x j 1 u ∂ x j 2 u , x , t ∈ R , where u is a real- (or complex-) valued function. We show that the associated initial value problem is well-posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when a 0 , k ≠ 0 for some k > j , in the sense that this equation cannot have its flow map C 2 at the origin in H s ( R ) , for any s ∈ R . The same technique leads to similar ill-posedness results for other higher-order nonlinear dispersive equation as higher-order Benjamin–Ono and intermediate long wave equations.

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

View Paper: https://www.semanticscholar.org/paper/2acd42061ce0eb7ae39e41da4b6c89e49953f042

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Well-posedness for the fifth-order KdV equation in the energy space (2012, methodology): “For a1, a2, a3, a4 ∈ R, we define the quantities amax ≥ asub ≥ athd ≥ amin to be 6This would be in contradiction with the C2-ill-posedness results in [32].”
The Cauchy problem for the Benjamin-Ono equation in $L^2$ revisited (2010, background): “9) fails to be smooth in any Sobolev space H(R), s ∈ R [24].”