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1.
Takahiro Okabe 《Journal of Differential Equations》2018,264(2):728-754
We consider the space-time behavior of the two dimensional Navier–Stokes flow. Introducing some qualitative structure of initial data, we succeed to derive the first order asymptotic expansion of the Navier–Stokes flow without moment condition on initial data in . Moreover, we characterize the necessary and sufficient condition for the rapid energy decay as motivated by Miyakawa–Schonbek [21]. By weighted estimated in Hardy spaces, we discuss the possibility of the second order asymptotic expansion of the Navier–Stokes flow assuming the first order moment condition on initial data. Moreover, observing that the Navier–Stokes flow lies in the Hardy space for , we consider the asymptotic expansions in terms of Hardy-norm. Finally we consider the rapid time decay as with cyclic symmetry introduced by Brandolese [2]. 相似文献
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In this paper, we consider the Cauchy problem for a two-phase model with magnetic field in three dimensions. The global existence and uniqueness of strong solution as well as the time decay estimates in are obtained by introducing a new linearized system with respect to for constants and , and doing some new a priori estimates in Sobolev Spaces to get the uniform upper bound of in norm. 相似文献
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M.M. Cavalcanti V.N. Domingos Cavalcanti M.A. Jorge Silva A.Y. de Souza Franco 《Journal of Differential Equations》2018,264(11):6535-6584
In this paper we discuss the asymptotic stability as well as the well-posedness of the damped wave equation posed on a bounded domain Ω of , subject to a locally distributed viscoelastic effect driven by a nonnegative function and supplemented with a frictional damping acting on a region A of Ω, where in A. Assuming that is constant, considering that the well-known geometric control condition holds and supposing that the relaxation function g is bounded by a function that decays exponentially to zero, we prove that the solutions to the corresponding partial viscoelastic model decay exponentially to zero, even in the absence of the frictional dissipative effect. In addition, in some suitable cases where the material density is not constant, it is also possible to remove the frictional damping term , that is, the localized viscoelastic damping is strong enough to assure that the system is exponentially stable. The semi-linear case is also considered. 相似文献
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For Toeplitz operators acting on the weighted Fock space , we consider the semi-commutator , where is a certain weight parameter that may be interpreted as Planck's constant ? in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit
()
It is well-known that tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to by Bauer and Coburn. We now further generalize (?) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra of bounded functions having vanishing mean oscillation on . Our approach is based on the algebraic identity , where denotes the Hankel operator corresponding to the symbol g, and norm estimates in terms of the (weighted) heat transform. As a consequence, only f (or likewise only g) has to be contained in one of the above classes for (?) to vanish. For g we only have to impose , e.g. . We prove that the set of all symbols with the property that for all coincides with . Additionally, we show that holds for all . Finally, we present new examples, including bounded smooth functions, where (?) does not vanish. 相似文献
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Tej-Eddine Ghoul Van Tien Nguyen Hatem Zaag 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2018,35(6):1577-1630
We consider the following parabolic system whose nonlinearity has no gradient structure: in the whole space , where and . We show the existence of initial data such that the corresponding solution to this system blows up in finite time simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics: with and . The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case ; and the fact that the case breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data. 相似文献
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Marie Doumic Miguel Escobedo Magali Tournus 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2018,35(7):1847-1884
We consider the fragmentation equation and address the question of estimating the fragmentation parameters – i.e. the division rate and the fragmentation kernel – from measurements of the size distribution at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance Xue and Radford (2013) [26] for amyloid fibril breakage. Under the assumption of a polynomial division rate and a self-similar fragmentation kernel , we use the asymptotic behavior proved in Escobedo et al. (2004) [11] to obtain uniqueness of the triplet and a representation formula for . To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral. 相似文献
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Daniela Giachetti Pedro J. Martínez-Aparicio François Murat 《Journal of Functional Analysis》2018,274(6):1747-1789
In the present paper we perform the homogenization of the semilinear elliptic problem In this problem is a Carathéodory function such that a.e. for every , with h in some and Γ a function such that and for every . On the other hand the open sets are obtained by removing many small holes from a fixed open set Ω in such a way that a “strange term” appears in the limit equation in the case where the function depends only on x.We already treated this problem in the case of a “mild singularity”, namely in the case where the function satisfies . In this case the solution to the problem belongs to and its definition is a “natural” and rather usual one.In the general case where exhibits a “strong singularity” at , which is the purpose of the present paper, the solution to the problem only belongs to but in general does not belong to anymore, even if vanishes on in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.In the present paper, using this definition, we perform the homogenization of the above semilinear problem and we prove that in the homogenized problem, the “strange term” still appears in the left-hand side while the source term is not modified in the right-hand side. 相似文献
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Gerd Grubb 《Journal of Functional Analysis》2018,274(9):2634-2660
This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations with a nonlocal operator P:
(*)
1) For strongly elliptic pseudodifferential operators (ψdo's) P on of order , a symbol calculus on is introduced that allows showing optimal regularity results, globally over and locally over : for , . The are anisotropic Sobolev spaces of Bessel-potential type, and there is a similar result for Besov spaces.2) Let Ω be smooth bounded, and let P equal (), or its generalizations to singular integral operators with regular kernels, generating stable Lévy processes. With the Dirichlet condition , the initial condition , and , (*) has a unique solution with . Here if , and is contained in if , but contains nontrivial elements from if (where ). The interior regularity of u is lifted when f is more smooth. 相似文献
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We consider the nonlinear Schrödinger equation in all dimensions , where and . We construct a class of initial values for which the corresponding solution is global and decays as , like if and like if . Moreover, we give an asymptotic expansion of those solutions as . We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at . To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents. 相似文献
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It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier–Stokes and MHD equations are Hölder continuous near boundary provided that either or is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points near the boundary is zero. This generalizes recent interior regularity results by Dong–Strain [5]. 相似文献
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Michael Winkler 《Journal of Differential Equations》2018,264(3):2310-2350
The chemotaxis system is considered under homogeneous Neumann boundary conditions in the ball , where and .Despite its great relevance as a model for the spontaneous emergence of spatial structures in populations of primitive bacteria, since its introduction by Keller and Segel in 1971 this system has been lacking a satisfactory theory even at the level of the basic questions from the context of well-posedness; global existence results in the literature are restricted to spatially one- or two-dimensional cases so far, or alternatively require certain smallness hypotheses on the initial data.For all suitably regular and radially symmetric initial data satisfying and , the present paper establishes the existence of a globally defined pair of radially symmetric functions which are continuous in and smooth in , and which solve the corresponding initial-boundary value problem for (?) with in an appropriate generalized sense. To the best of our knowledge, this in particular provides the first result on global existence for the three-dimensional version of (?) involving arbitrarily large initial data. 相似文献
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A.S. Sivatski 《Journal of Pure and Applied Algebra》2018,222(3):560-567
Let F be a field of characteristic distinct from 2, a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, , their matrices. We say that the pair is a k-pair if there exist such that all the entries of the upper-left corner of the matrices and are in F. We give certain criteria to determine whether a given pair is a k-pair. We consider the transfer determined by the -linear map with , , and prove that if , then is a -pair. If, additionally, the form does not have a totally isotropic subspace of dimension over , we show that is a -pair. In particular, if the form is anisotropic, and , then is a k-pair. 相似文献
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A.P. Bergamasco P.L. Dattori da Silva R.B. Gonzalez 《Journal of Differential Equations》2018,264(5):3500-3526
Let be a vector field defined on the torus , where , are real-valued functions and belonging to the Gevrey class , , for . We present a complete characterization for the s-global solvability and s-global hypoellipticity of L. Our results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets. 相似文献
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A. Jänig 《Journal of Differential Equations》2018,264(9):5713-5733
We consider asymptotically autonomous semilinear parabolic equations Suppose that as , where the semiflows induced by
(*)
are gradient-like. Under certain assumptions, it is shown that generically with respect to a perturbation g with as , every solution of is a connection between equilibria of (*) with . Moreover, if the Morse indices satisfy , then u is isolated by linearization. 相似文献
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In this paper we focus our attention on the following nonlinear fractional Schrödinger equation with magnetic field where is a parameter, , , is the fractional magnetic Laplacian, and are continuous potentials and is a subcritical nonlinearity. By applying variational methods and Ljusternick–Schnirelmann theory, we prove existence and multiplicity of solutions for ε small. 相似文献