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The Cauchy problem for the (generalized) Kadomtsev-Petviashvili-Burgers equation


Molinet, L (2000). The Cauchy problem for the (generalized) Kadomtsev-Petviashvili-Burgers equation. Differential and Integral Equations, 13(1-3):189-216.

Abstract

We investigate the Cauchy problem for the generalized Kadomtsev-Petviashvili-Burgers equation
u t +u xxx +u p u x +εv y -νu xx =0,v x =u y ,u(0)=ϕ
in Sobolev spaces. This nonlinear wave equation has both dispersive and dissipative parts. After showing local existence by the contraction principle for initial data ϕ∈H s (ℝ 2 ) such that ℱ -1 (k 2 k 1 ϕ ^)∈H r (ℝ 2 ), 0≤r≤s-1, we extend the solutions for all positive times. Whereas for ε=-1 and 1≤p<4/3 this is done without any assumption on the initial data, we require a smallness condition on the initial data otherwise. In a last part, we prove a local smoothing effect in the transverse direction, which enables us to establish the existence of weak global solutions in L 2 (ℝ 2 ) when ε=-1 and 1≤p<4/3.

Abstract

We investigate the Cauchy problem for the generalized Kadomtsev-Petviashvili-Burgers equation
u t +u xxx +u p u x +εv y -νu xx =0,v x =u y ,u(0)=ϕ
in Sobolev spaces. This nonlinear wave equation has both dispersive and dissipative parts. After showing local existence by the contraction principle for initial data ϕ∈H s (ℝ 2 ) such that ℱ -1 (k 2 k 1 ϕ ^)∈H r (ℝ 2 ), 0≤r≤s-1, we extend the solutions for all positive times. Whereas for ε=-1 and 1≤p<4/3 this is done without any assumption on the initial data, we require a smallness condition on the initial data otherwise. In a last part, we prove a local smoothing effect in the transverse direction, which enables us to establish the existence of weak global solutions in L 2 (ℝ 2 ) when ε=-1 and 1≤p<4/3.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:local existence; smallness condition on the initial data; local smoothing effect; existence of weak global solutions
Language:English
Date:2000
Deposited On:29 Nov 2010 16:27
Last Modified:05 Apr 2016 13:26
Publisher:Khayyam
ISSN:0893-4983
Official URL:http://www.aftabi.com/DIE/die13.html
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1811955
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0974.35109

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