2009/09/16 by Axel Gruenrock, Gruenrock, Axel
Mathematics · Physics and Astronomy · #35Q53 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.0909.2971
openalex publication_date 2009/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces Hrs(\R) defined by the norm \nv0Hrs(\R) := \nlt; ξgt; sv0Lr'ξ, lt; ξgt;=(1+ξ2)\frac12, (1)/(r)+(1)/(r')=1. Local well-posedness for the jth equation is shown in the parameter range 2 ≥ r >1, s ≥ (2j-1)/(2r'). The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C0-uniform sense, if s (2j)/(2j-1), independent of the size of s∈ \R. Especially for j≥ 2 we have C2-ill-posedness in Hs(\R). With similar arguments as used before in the mKdV context it is shown that this problem is locally well-posed in Hrs(\R), if 1 j - \frac32 - (1)/(2j) +(2j-1)/(2r'). For KdV itself the lower bound on s is pushed further down to s>max(-\frac12-(1)/(2r'),-\frac14-(11)/(8r')), where r∈ (1,2). These results rely on the contraction mapping principle, and the flow map is real analytic.