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1.
In this paper, we consider the stability to the global large solutions of 3-D incompressible Navier-Stokes equations in the anisotropic Sobolev spaces. In particular, we proved that for any , given a global large solution vC([0,∞);H0,s0(R3)∩L3(R3)) of (1.1) with and a divergence free vector satisfying for some sufficiently small constant depending on , v, and , (1.1) supplemented with initial data v(0)+w0 has a unique global solution in uC([0,∞);H0,s0(R3)) with ∇uL2(R+,H0,s0(R3)). Furthermore, uh is close enough to vh in C([0,∞);H0,s(R3)).  相似文献   

2.
Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤pq<, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v0(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality.  相似文献   

3.
In this paper we prove that in the general case (i.e. β not necessarily vanishing) the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed in , and if β=0 then it is locally well-posed in with . These results improve the corresponding results of Corcho and Linares (2007) [5]. Idea of the proof is to establish some bilinear and trilinear estimates in the space Gs×Fs, where Gs and Fs are dyadic Bourgain-type spaces related to the Schrödinger operator and the Airy operator , respectively, but with a modification on Fs in low frequency part of functions with a weaker structure related to the maximal function estimate of the Airy operator.  相似文献   

4.
5.
We establish C1,γ-partial regularity of minimizers of non-autono-mous convex integral functionals of the type: , with non-standard growth conditions into the gradient variable
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6.
We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Ω=Ω0×(0,L)∈R3. We show existence of a solution , p>3, where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of . The term u⋅∇w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (vn,ρn) that is bounded in and satisfies the Cauchy condition in a larger space L(0,L;L2(Ω0)) what enables us to deduce that the weak limit of a subsequence of (vn,ρn) is in fact a strong solution to our problem.  相似文献   

7.
We consider steady compressible Navier-Stokes-Fourier system for a gas with pressure p and internal energy e related by the constitutive law p=(γ−1)?e, γ>1. We show that for any there exists a variational entropy solution (i.e. solution satisfying the weak formulation of balance of mass and momentum, entropy inequality and global balance of total energy). This result includes the model for monoatomic gas (). If , these solutions also fulfill the weak formulation of the pointwise total energy balance.  相似文献   

8.
9.
There are only very few results on the existence of unique local in time strong solutions of the Navier-Stokes equations for completely general domains ΩR3, although domains with edges and corners, bounded or unbounded, are very important in applications. The reason is that the Lq-theory for the Stokes operator A is available in general only in the Hilbert space setting, i.e., with q=2. Our main result for a general domain Ω is optimal in a certain sense: Consider an initial value and a zero external force. Then the condition is sufficient and necessary for the existence of a unique local strong solution uL8(0,T;L4(Ω)) in some interval [0,T), 0<T, with u(0)=u0, satisfying Serrin’s condition . Note that Fujita-Kato’s sufficient condition u0D(A1/4) is strictly stronger and therefore not optimal.  相似文献   

10.
In this article we consider the following generalized quasi-geostrophic equation
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11.
We associate with any game G another game, which is a variant of it, and which we call . Winning strategies for have a lower recursive degree than winning strategies for G: if a player has a winning strategy of recursive degree 1 over G, then it has a recursive winning strategy over , and vice versa. Through we can express in algorithmic form, as a recursive winning strategy, many (but not all) common proofs of non-constructive Mathematics, namely exactly the theorems of the sub-classical logic Limit Computable Mathematics (Hayashi (2006) [6], Hayashi and Nakata (2001) [7]).  相似文献   

12.
In the first part of the paper, we prove the existence of a unique global solution to the axisymmetric Navier-Stokes system with initial data and external force with . This improves the result obtained by S. Leonardi, J. Málek, J. Necǎs and M. Pokorný [S. Leonardi, J. Málek, J. Necǎs, M. Pokorný, On axially symmetric flows in R3, Zeitschrift für analysis und ihre anwendungen, J. Anal. Appl. 18 (3) (1999) 639-649], where H2(R3) regularity was required. In the second part, we state global existence and uniqueness for the axisymmetric Navier-Stokes system with initial data in W2,p(R3) and external force in with 1<p<2. This also improves [S. Leonardi, J. Málek, J. Necǎs, M. Pokorný, On axially symmetric flows in R3, Zeitschrift für analysis und ihre anwendungen, J. Anal. Appl. 18 (3) (1999) 639-649] because less integrability is required on v0 and on f.  相似文献   

13.
14.
Let sR. In this paper, the authors first establish the maximal function characterizations of the Besov-type space with and τ∈[0,), the Triebel-Lizorkin-type space with p∈(0,), q∈(0,] and τ∈[0,), the Besov-Hausdorff space with p∈(1,), q∈[1,) and and the Triebel-Lizorkin-Hausdorff space with and , where t denotes the conjugate index of t∈[1,]. Using this characterization, the authors further obtain the local mean characterizations of these function spaces via functions satisfying the Tauberian condition and establish a Fourier multiplier theorem on these spaces. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking τ=0 and are also new even for Q spaces and Hardy-Hausdorff spaces.  相似文献   

15.
We take up the existence and global behavior of positive continuous solutions of the following nonlinear parabolic equation in (n?2) with boundary conditions u=0 on and u(x,0)=u0(x). The nonlinear term is required to satisfy some conditions related to a functional class , which we introduce in this paper and will be called parabolic Kato class in the half space. Our approach is based on potential theory.  相似文献   

16.
In this paper we prove the local and global well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces under certain conditions of s, q, and σ.  相似文献   

17.
18.
We consider the 2m-th order elliptic boundary value problem Lu=f(x,u) on a bounded smooth domain ΩRN with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic linear operator of order 2m whose principle part is of the form . We assume that f is superlinear at the origin and satisfies , , where are positive functions and q>1 is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution.  相似文献   

19.
20.
We consider the first initial boundary value problem for the non-autonomous nonclassical diffusion equation utεΔutΔu+f(u)=g(t), ε∈[0,1], in a bounded domain in RN. Under a Sobolev growth rate of the nonlinearity f and a suitable exponential growth of the external force g, using the asymptotic a priori estimate method, we prove the existence of pullback D-attractors in the space and the upper semicontinuity of at ε=0.  相似文献   

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