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1.
Xingyu Yang 《Journal of Differential Equations》2010,248(6):1458-1472
The Cauchy problem of a fifth-order shallow water equation
2.
In this paper we mainly study the Cauchy problem for the generalized shallow water wave equation in the Sobolev space Hs of lower order s. Using the crucial bilinear estimates in the Fourier transform restriction spaces related to the shallow water wave equation, we establish local well-posedness in Hs with any . 相似文献
3.
In this paper we establish a new bilinear estimate in suitable Bourgain spaces by using a fundamental estimate on dyadic blocks for the Kawahara equation which was obtained by the [k;Z] multiplier norm method of Tao (2001) [2]; then the local well-posedness of the Cauchy problem for a fifth-order shallow water wave equation in with is obtained by the Fourier restriction norm method. And some ill-posedness in with is derived from a general principle of Bejenaru and Tao. 相似文献
4.
Yueling Jia 《Journal of Differential Equations》2009,246(6):2448-405
The Cauchy problems for some kind of fifth-order shallow water equations
5.
Yongsheng Li 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(6):1610-1625
In this paper we consider the initial value problem of the Benjamin equation
6.
Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation
7.
Soonsik Kwon 《Journal of Differential Equations》2008,245(9):2627-2659
In this paper we prove that the following fifth-order equation arising from the KdV hierarchy
8.
Feride Tiğlay 《Journal of Evolution Equations》2005,5(4):509-527
We prove that the periodic initial value problem for the modified Hunter-Saxton equation is locally well-posed for initial
data in the space of continuously differentiable functions on the circle and in Sobolev spaces
when s > 3/2. We also study the analytic regularity (both in space and time variables) of this problem and prove a Cauchy-Kowalevski
type theorem. Our approach is to rewrite the equation and derive the estimates which permit application of o.d.e. techniques
in Banach spaces. For the analytic regularity we use a contraction argument on an appropriate scale of Banach spaces to obtain
analyticity in both time and space variables. 相似文献
9.
We make use of the method of modulus of continuity [A. Kiselev, F. Nazarov, R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 (2008) 211-240] and Fourier localization technique [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] to prove the global well-posedness of the critical Burgers equation t∂u+ux∂u+Λu=0 in critical Besov spaces with p∈[1,∞), where . 相似文献
10.
We prove the global well-posedness of the KP-I equation in for data of arbitrary size in a suitable Sobolev class. We use a compactness method to get local solutions. We extend the
solutions thanks to some conservation laws, an almost conserved quantity and Strichartz estimates.
Received: 5 April 2001 / Revised version: 22 October 2001 / Published online: 17 June 2002 相似文献
11.
Having the ill-posedness in the range s<−3/4 of the Cauchy problem for the Benjamin equation with an initial Hs(R) data, we prove that the already-established local well-posedness in the range s>−3/4 of this initial value problem is extendable to s=−3/4 and also that such a well-posed property is globally valid for s∈[−3/4,∞). 相似文献
12.
David Karapetyan 《Journal of Differential Equations》2010,249(4):796-826
It is shown that the solution map for the hyperelastic rod equation is not uniformly continuous on bounded sets of Sobolev spaces with exponent greater than 3/2 in the periodic case and greater than 1 in the non-periodic case. The proof is based on the method of approximate solutions and well-posedness estimates for the solution and its lifespan. 相似文献
13.
Pedro Isaza 《Journal of Differential Equations》2006,230(2):661-681
In this article we consider the initial value problem for the Ostrovsky equation:
14.
The local well-posedness for the generalized two-dimensional (2D) Ginzburg-Landau equation is obtained for initial data in Hs(R2)(s>1/2). The global result is also obtained in Hs(R2)(s>1/2) under some conditions. The results on local and global well-posedness are sharp except the endpoint s=1/2. We mainly use the Tao's [k;Z]-multiplier method to obtain the trilinear and multilinear estimates. 相似文献
15.
In this paper, we study the continuation of solutions to an equation for surface water waves of moderate amplitude in the shallow water regime beyond wave breaking (in [11], Constantin and Lannes proved that this equation accommodates wave breaking phenomena). Our approach is based on a method proposed by Bressan and Constantin [2]. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, the evolution problem is rewritten as a semilinear system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows one to continue the solution after collision time, giving a global conservative solution where the energy is conserved for almost all times. Finally, returning to the original variables, we obtain a semigroup of global conservative solutions, which depend continuously on the initial data. 相似文献
16.
L.M. Bragança 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(9):2991-3003
We investigate some well-posedness issues for the initial value problem (IVP) associated with the system
17.
In this paper we prove that the already-established local well-posedness in the range s>−5/4 of the Cauchy problem with an initial Hs(R) data for a fifth-order shallow water wave equation is extendable to s=−5/4 by using the space. This is sharp in the sense that the ill-posedness in the range s<−5/4 of this initial value problem is already known. 相似文献
18.
19.
This paper is mainly concerned with the periodic Cauchy problem for a generalized two-component μ-Hunter-Saxton system with analytic initial data. The analyticity of its solutions is proved in both variables, globally in space and locally in time. The obtained result can be also applied to its special cases—the classical integrable two-component Hunter-Saxton system, the generalized μ-Hunter-Saxton equation and the classical Hunter-Saxton equation. 相似文献