共查询到20条相似文献,搜索用时 15 毫秒
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In this paper, we study the optimal decay rates of solutions for the generalized Benjamin–Bona–Mahony equation in multi-dimensional space (n≥3). By using Fourier transform and the energy method, we obtain the Lq(2≤q≤∞) convergence rates of the solutions under the condition that the initial data is small. The optimal decay rates obtained in this paper are found to be the same as the decay rate for the Heat equation. 相似文献
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We investigate the Cauchy problem and the initial-boundary value problem for multi-dimensional conservation laws with degenerate viscosity in the whole space and in the half-space respectively. We give the optimal decay estimates in the W1,p(1≤p≤∞) norm for the perturbation from the planar viscous rarefaction wave. The analysis based on the new Lp-energy method and L1-estimates. 相似文献
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We consider nonlinear elliptic equations of p -Laplacian type that are not necessarily of variation form when the nonlinearity is allowed to be discontinuous and the boundary of the domain can go beyond the Lipschitz category. Under smallness in the BMO nonlinearity and sufficient flatness of the Reifenberg domain, we obtain the global weighted Lq estimates with q∈(p,∞) for the gradient of weak solutions. 相似文献
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We study the best decay rate of the solutions of a damped Euler–Bernoulli beam equation with a homogeneous Dirichlet boundary conditions. We show that the fastest decay rate is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup, if the damping coefficient is in L∞(0,1). 相似文献
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Clayton Bjorland Maria E. Schonbek 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2008
We consider the viscous n -dimensional Camassa–Holm equations, with n=2,3,4 in the whole space. We establish existence and regularity of the solutions and study the large time behavior of the solutions in several Sobolev spaces. We first show that if the data is only in L2 then the solution decays without a rate and that this is the best that can be expected for data in L2. For solutions with data in Hm∩L1 we obtain decay at an algebraic rate which is optimal in the sense that it coincides with the rate of the underlying linear part. 相似文献
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We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical (α<1/2) dissipation α(−Δ). This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical (α=1/2) QG equation [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray–Hopf weak solutions: from L2 to L∞, from L∞ to Hölder (Cδ, δ>0), and from Hölder to classical solutions. In the supercritical case, Leray–Hopf weak solutions can still be shown to be L∞, but it does not appear that their approach can be easily extended to establish the Hölder continuity of L∞ solutions. In order for their approach to work, we require the velocity to be in the Hölder space C1−2α. Higher regularity starting from Cδ with δ>1−2α can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. 相似文献
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This paper is concerned with special regularity properties of the solutions to the Maxwell–Landau–Lifshitz (MLL) system describing ferromagnetic medium. Besides the classical results on the boundedness of ∂tm,∂tE and ∂tH in the spaces L∞(I,L2(Ω)) and L2(I,W1,2(Ω)) we derive also estimates in weighted Sobolev spaces. This kind of estimates can be used to control the Taylor remainder when estimating the error of a numerical scheme. 相似文献
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In this paper we prove existence of global solutions and (L2(Ω)×L2(Γ),(H1(Ω)∩Lp(Ω))×Lp(Γ))-global attractors for semilinear parabolic equations with dynamic boundary conditions in bounded domains with a smooth boundary, where there is no other restriction on p(≥2). 相似文献
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This paper is concerned with the existence, uniqueness, and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier–Stokes–Korteweg system effected by the given mass source, the external force of general form, and the energy source in R3. Based on the weighted L2-method and some delicate L∞ estimates on solutions to the linearized problem, the existence and uniqueness of stationary solution are obtained by the contraction mapping principle. The proof of the stability result is given by an elementary energy method and relies on some intrinsic properties of the full compressible Navier–Stokes–Korteweg system. 相似文献
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We consider two-dimensional mixed problems in an exterior domain for a semilinear strongly damped wave equation with a power-type nonlinearity |u|p. If the initial data have a small weighted energy, we shall derive a global existence and energy decay results in the case when the power p of the nonlinear term satisfies p>6. 相似文献
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In the Hammersley harness processes the R-valued height at each site i∈Zd is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in Zd. With this representation we compute covariances and show L2 and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L2 at speed t1−d/2 (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process. 相似文献
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Nathaël Alibaud Boris Andreianov 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2010
The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional Laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the L∞-framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail. 相似文献
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This paper is concerned with the existence of a global attractor for a semiflow governed by the weak solutions to a nonlinear one-dimensional thermoviscoelasticity with a non-convex free energy density. The constitutive assumptions for the Helmholtz free energy include the model for the study of martensitic phase transitions in shape memory alloys. To describe physically phase transitions between different configurations of crystal lattices, we work in a framework in which the strain u belongs to L∞. New approaches are introduced and more delicate estimates are derived to establish the crucial L∞-estimate of strain u in deriving the compactness of the orbit of the semiflow and the existence of an absorbing set. 相似文献