共查询到20条相似文献,搜索用时 31 毫秒
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Pedro Isaza 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(11):4016-4029
In this paper we prove that sufficiently smooth solutions of the Ostrovsky equation with positive dispersion,
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Pedro Isaza 《Journal of Differential Equations》2009,247(6):1851-4029
In this article we prove that sufficiently smooth solutions of the Ostrovsky equation with negative dispersion:
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In this article we prove that sufficiently smooth solutions of the Zakharov-Kuznetsov equation:
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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
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By using the I-method, we prove that the Cauchy problem of the fifth-order shallow water equation is globally well-posed in the Sobolev space Hs(R) provided . 相似文献
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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
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Songzhe Lian 《Journal of Differential Equations》2008,244(5):1178-1209
For , the author studies the existence of a kind of weak solution to the Cauchy problem
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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
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Mariano De Leo 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(4):979-986
This paper is concerned with the existence of ground states for the Schrödinger-Poisson equation , where V(u) is a Hartree type nonlinearity, stemming from the coupling with the Poisson equation, which includes the so-called doping profile or impurities. By means of variational methods in the energy space we show that ground states exist and belong to the Schwartz space of rapidly decreasing functions whenever total charge not exceed some critical value, it is also shown that for values of the total charge greater than this critical value, energy is not bounded from below. In addition, we show that this critical value is the total charge given by the impurities. 相似文献
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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
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Wengu Chen 《Journal of Differential Equations》2008,245(11):3433-3469
We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as , while α∈R. We introduce an interpolated energy space Es to consider the well-posedness of the initial value problem (IVP) of the fifth order KP-I equation. We obtain the local well-posedness of IVP of the fifth order KP-I equation in Es for 0<s?1. To obtain the local well-posedness, we present a bilinear estimate in the Bourgain space in the framework of the interpolated energy space. It crucially depends on the dyadic decomposed Strichartz estimate, the fifth order dispersive smoothing effect and maximal estimate. 相似文献
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We consider a dissipative version of the modified Korteweg-de Vries equation ut+uxxx−uxx+x(u3)=0. We prove global well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s>−1/4 while for s<−1/2 we prove some ill-posedness issues. 相似文献
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Vladimir Varlamov 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(6):957-985
The Ostrovsky equation governs the propagation of long nonlinear surface waves in the presence of rotation. It is related
to the Korteweg-de Vries (KdV) and the Kadomtsev-Petviashvili models. KdV can be obtained from the equation in question when
the rotation parameter γ equals zero. A fundamental solution of the Cauchy problem for the linear Ostrovsky equation is presented
in the form of an oscillatory Fourier integral. Another integral representation involving Airy and Bessel functions is derived
for it. It is shown that its asymptotic expansion as γ → 0 contains the KdV fundamental solution as the zero term. The Airy
transform is used to establish some of its properties. Higher-order asymptotics for γ → 0 on a bounded time interval are obtained
for both the fundamental solution and the solution of the linear Cauchy problem for the Ostrovsky equation.
Received: November 23, 2004; revised: March 13, 2005
Research is supported by US Department of Defense, under grant No. DAAD19-03-1-0204 相似文献
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In this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in “critical” Sobolev spaces. Moreover, the Gevrey class that we obtain is “near optimal” and as a corollary, we obtain temporal decay rates of higher order Sobolev norms of the solutions. Unlike the Navier–Stokes or the subcritical quasi-geostrophic equations, the low dissipation poses a difficulty in establishing Gevrey regularity. A new commutator estimate in Gevrey classes, involving the dyadic Littlewood–Paley operators, is established that allow us to exploit the cancellation properties of the equation and circumvent this difficulty. 相似文献