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1.
Let H denote the halfline [0,∞). A point p?βH?H is called a near point if p is in the closure of some countable discrete closed subspace of H. In addition, a point p?βH?H is called a large point if p is not in the closure of a closed subset of H of finite Lebesgue measure. We will show that for every autohomeomorphism ? of βH?H and for each near point p we have that ?(p) is not large. In addition, we establish, under CH, the existence of a point x?βH?H such that for each autohomeomorphism ? of βH?H the point ?(x) is neither large nor near. 相似文献
2.
This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid in two-dimensional (2D) bounded domains. We first prove the existence of pullback attractors AV in space V (has H2-regularity, see notation in Section 2) and AH in space H (has L2-regularity) for the cocycle corresponding to the solutions of the fluid. Then we verify the regularity of the pullback attractors by showing AV=AH, which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data. 相似文献
3.
We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction orthogonal to H. For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution u is symmetric about H . Moreover, we prove that if u?0, then the nodal set of u divides the domain Ω into a finite number of reflectionally symmetric subdomains in which u has the usual Gidas–Ni–Nirenberg symmetry and monotonicity properties. We also show several examples of nonnegative solutions with a nonempty interior nodal set. 相似文献
4.
Thierry Cazenave Daoyuan Fang 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2011,28(1):135
For the nonlinear Schrödinger equation iut+Δu+λα|u|u=0 in RN, local existence of solutions in Hs is well known in the Hs-subcritical and critical cases 0<α?4/(N−2s), where 0<s<min{N/2,1}. However, even though the solution is constructed by a fixed-point technique, continuous dependence in Hs does not follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial value in the sense that the local flow is continuous Hs→Hs. If, in addition, α?1 then the flow is locally Lipschitz. 相似文献
5.
Goro Akagi 《Journal of Differential Equations》2006,231(1):32-56
We prove the existence of solutions of the Cauchy problem for the doubly nonlinear evolution equation: dv(t)/dt+V∂φt(u(t))∋f(t), v(t)∈H∂ψ(u(t)), 0<t<T, where H∂ψ (respectively, V∂φt) denotes the subdifferential operator of a proper lower semicontinuous functional ψ (respectively, φt explicitly depending on t) from a Hilbert space H (respectively, reflexive Banach space V) into (−∞,+∞] and f is given. To do so, we suppose that V?H≡H∗?V∗ compactly and densely, and we also assume smoothness in t, boundedness and coercivity of φt in an appropriate sense, but use neither strong monotonicity nor boundedness of H∂ψ. The method of our proof relies on approximation problems in H and a couple of energy inequalities. We also treat the initial-boundary value problem of a non-autonomous degenerate elliptic-parabolic problem. 相似文献
6.
Eugene Tsyganov 《Journal of Differential Equations》2008,245(12):3936-3955
We construct global weak solution of the Navier-Stokes equations with capillarity and nonmonotonic pressure. The volume variable v0 is initially assumed to be in H1 and the velocity variable u0 to be in L2 on a finite interval [0,1]. We show that both variables become smooth in positive time and that asymptotically in time u→0 strongly in L2([0,1]) and v approaches the set of stationary solutions in H1([0,1]). 相似文献
7.
In this paper, we investigate an initial boundary value problem for 1D compressible isentropic Navier-Stokes equations with large initial data, density-dependent viscosity, external force, and vacuum. Making full use of the local estimates of the solutions in Cho and Kim (2006) [3] and the one-dimensional properties of the equations and the Sobolev inequalities, we get a unique global classical solution (ρ,u) where ρ∈C1([0,T];H1([0,1])) and u∈H1([0,T];H2([0,1])) for any T>0. As it is pointed out in Xin (1998) [31] that the smooth solution (ρ,u)∈C1([0,T];H3(R1)) (T is large enough) of the Cauchy problem must blow up in finite time when the initial density is of nontrivial compact support. It seems that the regularities of the solutions we obtained can be improved, which motivates us to obtain some new estimates with the help of a new test function ρ2utt, such as Lemmas 3.2-3.6. This leads to further regularities of (ρ,u) where ρ∈C1([0,T];H3([0,1])), u∈H1([0,T];H3([0,1])). It is still open whether the regularity of u could be improved to C1([0,T];H3([0,1])) with the appearance of vacuum, since it is not obvious that the solutions in C1([0,T];H3([0,1])) to the initial boundary value problem must blow up in finite time. 相似文献
8.
In this paper, we consider the defocusing, energy-critical Hartree equation with harmonic potential for the radial data in all dimensions (n≥5) and show the global well-posedness and scattering theory in the space Σ=H1∩FH1. We take advantage of some symmetry of the Hartree nonlinearity to exploit the derivative-like properties of the Galilean operators and obtain the energy control as well. Based on Bourgain and Tao’s approach, we use a localized Morawetz identity to show the global well-posedness. A key decay estimate comes from the linear part of the energy rather than the nonlinear part, which finally helps us to complete the scattering theory. 相似文献
9.
Shihui Zhu Jian ZhangHan Yang 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(17):6244-6255
This paper is concerned with the Cauchy problem for the biharmonic nonlinear Schrödinger equation with L2-super-critical nonlinearity. By establishing the profile decomposition of bounded sequences in H2(RN), the best constant of a Gagliardo-Nirenberg inequality is obtained. Moreover, a sufficient condition for the global existence of the solution to the biharmonic nonlinear Schrödinger equation is given. 相似文献
10.
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
11.
Amin Esfahani 《Journal of Differential Equations》2009,247(12):3181-323
In this paper we study the generalized BO-ZK equation in two space dimensions
ut+upux+αHuxx+εuxyy=0. 相似文献
12.
A generalization of the Camassa-Holm equation, a model for shallow water waves, is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs(R) with is established via a limiting procedure. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs with is developed. 相似文献
13.
Kotaro Tsugawa 《Journal of Differential Equations》2009,247(12):3163-3815
We consider the Cauchy problem of the Ostrovsky equation. We first prove the time local well-posedness in the anisotropic Sobolev space Hs,a with s>−a/2−3/4 and 0?a?−1 by the Fourier restriction norm method. This result include the time local well-posedness in Hs with s>−3/4 for both positive and negative dissipation, namely for both βγ>0 and βγ<0. We next consider the weak rotation limit. We prove that the solution of the Ostrovsky equation converges to the solution of the KdV equation when the rotation parameter γ goes to 0 and the initial data of the KdV equation is in L2. To show this result, we prove a bilinear estimate which is uniform with respect to γ. 相似文献
14.
We investigate the large-time behavior of classical solutions to the thin-film type equation ut=−x(uuxxx). It was shown in previous work of Carrillo and Toscani that for non-negative initial data u0 that belongs to H1(R) and also has a finite mass and second moment, the strong solutions relax in the L1(R) norm at an explicit rate to the unique self-similar source type solution with the same mass. The equation itself is gradient flow for an energy functional that controls the H1(R) norm, and so it is natural to expect that one should also have convergence in this norm. Carrillo and Toscani raised this question, but their methods, using a different Lyapunov functions that arises in the theory of the porous medium equation, do not directly address this since their Lyapunov functional does not involve derivatives of u. Here we show that the solutions do indeed converge in the H1(R) norm at an explicit, but slow, rate. The key to establishing this convergence is an asymptotic equipartition of the excess energy. Roughly speaking, the energy functional whose dissipation drives the evolution through gradient flow consists of two parts: one involving derivatives of u, and one that does not. We show that these must decay at related rates—due to the asymptotic equipartition—and then use the results of Carrillo and Toscani to control the rate for the part that does not depend on derivatives. From this, one gets a rate on the dissipation for all of the excess energy. 相似文献
15.
Indranil Biswas 《Bulletin des Sciences Mathématiques》2008,132(3):246
We give a complete classification of isomorphism classes of all SU(2)-equivariant holomorphic Hermitian vector bundles on CP1. We construct a canonical bijective correspondence between the isomorphism classes of SU(2)-equivariant holomorphic Hermitian vector bundles on CP1 and the isomorphism classes of pairs ({Hn}n∈Z,T), where each Hn is a finite dimensional Hilbert space with Hn=0 for all but finitely many n, and T is a linear operator on the direct sum ⊕n∈ZHn such that T(Hn)⊂Hn+2 for all n. 相似文献
16.
Romain Joly 《Journal of Differential Equations》2006,229(2):588-653
In this paper, we study the convergence of the wave equation with variable internal damping term γn(x)ut to the wave equation with boundary damping γ(x)⊗δx∈∂Ωut when (γn(x)) converges to γ(x)⊗δx∈∂Ω in the sense of distributions. When the domain Ω in which these equations are defined is an interval in R, we show that, under natural hypotheses, the compact global attractor of the wave equation damped on the interior converges in X=H1(Ω)×L2(Ω) to the one of the wave equation damped on the boundary, and that the dynamics on these attractors are equivalent. We also prove, in the higher-dimensional case, that the attractors are lower-semicontinuous in X and upper-semicontinuous in H1−ε(Ω)×H−ε(Ω). 相似文献
17.
A nonlinear shallow water equation, which includes the famous Camassa-Holm (CH) and Degasperis-Procesi (DP) equations as special cases, is investigated. The local well-posedness of solutions for the nonlinear equation in the Sobolev space Hs(R) with is developed. Provided that does not change sign, u0∈Hs () and u0∈L1(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x)∈C([0,∞);Hs(R))∩C1([0,∞);Hs−1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired. 相似文献
18.
In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in R3. By making use of Bourgain's method in conjunction with a precise Strichartz estimate of S. Klainerman and D. Tataru, we establish the Hs (s<1) global well-posedness of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the previously discussed conclusion, we obtain global solution for this non-scaling equation with small initial data in Hs0×Hs0−1 where but not , for this equation that we consider is a subconformal equation in some sense. In doing so a number of nonlinear prior estimates are already established by using Bony's decomposition, flexibility of Klein-Gordon admissible pairs which are slightly different from that of wave equation and a commutator estimate. We establish this commutator estimate by exploiting cancellation property and utilizing Coifman and Meyer multilinear multiplier theorem. As far as we know, it seems that this is the first result on low regularity for this Klein-Gordon-Hartree equation. 相似文献
19.
Jong Uhn Kim 《Journal of Differential Equations》2006,228(2):737-768
We discuss the Cauchy problem for the stochastic Benjamin-Ono equation in the function class Hs(R), s>3/2. When there is a zero-order dissipation, we also establish the existence of an invariant measure with support in H2(R). Many authors have discussed the Cauchy problem for the deterministic Benjamin-Ono equation. But our results are new for the stochastic Benjamin-Ono equation. Our goal is to extend known results for the deterministic equation to the stochastic equation. 相似文献
20.
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