A bilinear Airy-estimate with application to gKdV-3
By Axel Grunrock. Differential and Integral Equations, 2001
61 citations recorded by the source corpus.
The Fourier restriction norm method is used to show local wellposedness for the Cauchy-Problem \[u_t + u_{xxx} + (u^4)_x=0,\hspace{1cm}u(0)=u_0 \in H^s_x({\bf R}), \,\,\,s>-\tfrac{1}{6}\] for the generalized Korteweg-deVries equation of order three, for short gKdV-3. For real-valued data $u_0 \in L^2_x({\bf R})$ global wellposedness follows by the conservation of the $L^2$ norm. The main new tool is a bilinear estimate for solutions of the Airy-equation.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/25a51592b4f4e5c8d8028b673ad59839df99c590
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In Applied Mathematics
- Didier Pilod
- Carlos E. Kenig
- Aynur Bulut
- Gunther Uhlmann
- Raphael Cote
- B. Dodson
- Zhongwei Shen
- Luc Molinet
In Mathematical Physics
- Didier Pilod
- Carlos E. Kenig
- Aynur Bulut
- Raphael Cote
- Gunther Uhlmann
- B. Dodson
- Luc Molinet
- F. Merle
Citations in the corpus
- The Fourier restriction norm method for the Zakharov-Kuznetsov equation (2013, unknown): “Similar to [8] let I x,− be the bilinear operator with symbol |ξ1 − ξ2|, I x,+ be the bilinear operator with symbol |ξ1 + 2ξ2|, I y,− be the bilinear operator with symbol |η1 − η2| and I y,+ be the bilinear operator with symbol |η1 + 2η2|.”
- Well-posedness for the fifth-order KdV equation in the energy space (2012, methodology): “Next, we derive a bilinear Strichartz estimate for the group {et∂5 x}, which is an extension of the one proved in [10] for the Airy equation (see also Lemma 3.”