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Sharp well-posedness for a coupled system of mKdV-type equations

Sharp well-posedness for a coupled system of mKdV-type equations We consider the initial value problem associated with a system consisting modified Korteweg–de Vries-type equations $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tv + \partial _x^3v + \partial _x(vw^2) =0,&{}\quad u(x,0)=\phi (x),\\ \partial _tw + \alpha \partial _x^3w + \partial _x(v^2w) =0,&{}\quad v(x,0)=\psi (x), \end{array}\right. } \end{aligned}$$ ∂ t v + ∂ x 3 v + ∂ x ( v w 2 ) = 0 , u ( x , 0 ) = ϕ ( x ) , ∂ t w + α ∂ x 3 w + ∂ x ( v 2 w ) = 0 , v ( x , 0 ) = ψ ( x ) , and prove the local well-posedness results for given data in low regularity Sobolev spaces $$H^{s}({\mathbb {R}})\times H^{s}({\mathbb {R}})$$ H s ( R ) × H s ( R ) , $$s> -\,\frac{1}{2}$$ s > - 1 2 , for $$0<\alpha <1$$ 0 < α < 1 . Our result covers the whole scaling subcritical range of Sobolev regularity contrary to the case $$\alpha =1$$ α = 1 , where the local well-posedness holds only for $$s\ge \frac{1}{4}$$ s ≥ 1 4 . We also prove that the local well-posedness result is sharp in two different ways; namely, for $$s<-\,\frac{1}{2}$$ s < - 1 2 the key trilinear estimates used in the proof of the local well-posedness theorem fail to hold, and the flow map that takes initial data to the solution fails to be $$C^3$$ C 3 at the origin. These results hold for $$\alpha >1$$ α > 1 as well. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Sharp well-posedness for a coupled system of mKdV-type equations

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References (33)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-019-00508-6
Publisher site
See Article on Publisher Site

Abstract

We consider the initial value problem associated with a system consisting modified Korteweg–de Vries-type equations $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tv + \partial _x^3v + \partial _x(vw^2) =0,&{}\quad u(x,0)=\phi (x),\\ \partial _tw + \alpha \partial _x^3w + \partial _x(v^2w) =0,&{}\quad v(x,0)=\psi (x), \end{array}\right. } \end{aligned}$$ ∂ t v + ∂ x 3 v + ∂ x ( v w 2 ) = 0 , u ( x , 0 ) = ϕ ( x ) , ∂ t w + α ∂ x 3 w + ∂ x ( v 2 w ) = 0 , v ( x , 0 ) = ψ ( x ) , and prove the local well-posedness results for given data in low regularity Sobolev spaces $$H^{s}({\mathbb {R}})\times H^{s}({\mathbb {R}})$$ H s ( R ) × H s ( R ) , $$s> -\,\frac{1}{2}$$ s > - 1 2 , for $$0<\alpha <1$$ 0 < α < 1 . Our result covers the whole scaling subcritical range of Sobolev regularity contrary to the case $$\alpha =1$$ α = 1 , where the local well-posedness holds only for $$s\ge \frac{1}{4}$$ s ≥ 1 4 . We also prove that the local well-posedness result is sharp in two different ways; namely, for $$s<-\,\frac{1}{2}$$ s < - 1 2 the key trilinear estimates used in the proof of the local well-posedness theorem fail to hold, and the flow map that takes initial data to the solution fails to be $$C^3$$ C 3 at the origin. These results hold for $$\alpha >1$$ α > 1 as well.

Journal

Journal of Evolution Equations – Springer Journals

Published: Apr 29, 2019

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