Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4590927 | Journal of Functional Analysis | 2011 | 26 Pages |
Abstract
This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov–Kuznetsov equation, namely,{ut+∂xΔu+ukux=0,(x,y)∈R2,t>0,u(x,y,0)=u0(x,y). For 2⩽k⩽72⩽k⩽7, the IVP above is shown to be locally well posed for data in Hs(R2)Hs(R2), s>3/4s>3/4. For k⩾8k⩾8, local well-posedness is shown to hold for data in Hs(R2)Hs(R2), s>sks>sk, where sk=1−3/(2k−4)sk=1−3/(2k−4). Furthermore, for k⩾3k⩾3, if u0∈H1(R2)u0∈H1(R2) and satisfies ‖u0‖H1≪1‖u0‖H1≪1, then the solution is shown to be global in H1(R2)H1(R2). For k=2k=2, if u0∈Hs(R2)u0∈Hs(R2), s>53/63s>53/63, and satisfies ‖u0‖L2<3‖φ‖L2, where φ is the corresponding ground state solution, then the solution is shown to be global in Hs(R2)Hs(R2).
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Felipe Linares, Ademir Pastor,