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1.
§1. Introduction In this paper we consider the continuous dependence of the solutions of the Cauchy problem for Navier-Stokes equations on theirs initial data in various function spaces, the Nikol'ski space H_2~(1+l) and Ba space. The Ba space was introduced by Ding Xiaxi and Into Peizhu, as follows  相似文献   

2.
The author studies the Cauchy problem of the dissipative quasi-geostrophic equation in weak Morrey spaces. The global well-posedness is established for any small initial data in the weak space Mp^*,γ(R^n), with 1〈p〈∞and A = n-(2α-1)p, and for a small external force in a time-weighted weak Morrey space.  相似文献   

3.
We consider the Cauchy problem for a family of SchrSdinger equations with initial data in modulation spaces Mp,1^s. We develop the existence, uniqueness, blowup criterion, stability of regularity, scattering theory, and stability theory.  相似文献   

4.
To study the local regularity of solutions to second order elliptic partial differential equations, Morrey in [1] introduced some function spaces, which are called the Morrey spaces todaySince then, many mathematicians have studied regularities of solutions to some kinds of secondorder elliptic equations in Morrey spaces.We give the deceptions of Morrey space and weak Morrey space first.Definition 1 Let TheThesubset of those functions of Lp for which will be called the Morrey space LpDefi…  相似文献   

5.
In this paper we consider the Cauchy problems of Burgers' equations and the Deybe system. Their existence and uniqueness of the time-global solutions for small initial data in some pseudomeasure spaces are obtained. The asymptotic stability of small solutions is proved. As an immediate result the existence and uniqueness of the self-similar solutions are also obtained provided the initial data satisfy the self-similar structures.  相似文献   

6.
In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations withweak data n the homogeneous spaces.We give a method which can be used to construct local mild solutionsof the abstract Cauchy problem in C(σ,s,p)and L~q([O,T);H~(s,p)by introducing the concept of both admissiblequintuptet and compatible space and establishing estblishing time-space estimates for solutions to the linear parabolic typeequations For the small data,we prove that these results can be extended globally in time. We also study the  相似文献   

7.
This paper is a continue work of [4, 5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term -μ/((1+t)λ)ρu, where λ≥ 0 and μ 0 are constants. We have showed that, for all λ≥ 0 and μ 0, the smooth solution to the Cauchy problem exists globally or blows up in finite time. In the present paper, instead of the Cauchy problem we consider the initialboundary value problem in the half space R_+~d with space dimension d = 2, 3. With the help of the special structure of the equations and the fluid vorticity, we overcome the difficulty arisen from the boundary effect. We prove that there exists a global smooth solution for 0 ≤λ 1when the initial data is close to its equilibrium state. In addition, exponential decay of the fluid vorticity will also be established.  相似文献   

8.
We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in the three space dimensions with general initial data which could be either vacuum or non-vacuum under the assumption that the viscosity coefficient μ is large enough.  相似文献   

9.
We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R2. By exploiting the global-in-time estimate to the two-dimensional(2D for short) classical incompressible Navier-Stokes equations and using techniques developed in(SIAM J Math Anal, 2020, 52(2): 1806–1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity...  相似文献   

10.
In this paper we study the global existence and uniqueness of classical solutions to the Cauchy problem for 3D isentropic compressible Navier-Stokes equations with general initial data which could contain vacuum.We give the relation between the viscosity coefficients and the initial energy,which implies that the Cauchy problem under consideration has a global classical solution.  相似文献   

11.
The aim of this work is to investigate the integrability properties of the maximal operator Mu,associated with a non-doubling measure μ defined on Rn. We start by establishing for a wide class of radial and increasing measures μ that Mu is bounded on all the spaces Lu^p(R^n),P〉1.Also,we show that there is a radial and increasing measure p for which Mμ does not map Lμ^p(R^n) into weak Lμ^p(R^n),1≤p〈∞.  相似文献   

12.
It is proved that a class of multilinear singular integral operators with homogeneous kernels are bounded operators from the product spaces to the Hardy spacesH r , (ℝ n ) and the weak Hardy spaceH r,∞ (ℝ n . As an application of this result, the L p ,(ℝ n ) boundedness of a class of commutator for the singular integral with homogeneous kernels is obtained. Project supparted in part by the National Natural Science Foundation of Chind (Grant No. 19131080) of China and Doctoral Programme Foundation of Institution of Higher Education (Grant No. 98002703) of China.  相似文献   

13.
In this paper, the authors give a characterization of the (L p, λ , L q, λ )-compactness for the Riesz potential commutator [b,I α ]. More precisely, the authors prove that the commutator [b,I α ] is a compact operator from the Morrey space L p, λ (ℝ n ) to L q, λ (ℝ n ) if and only if b ∈ VMO(ℝ n ), the BMO-closure of . The research was supported by NSF of China (Grant: 10571015, 10826046) and SRFDP of China (Grant: 20050027025).  相似文献   

14.
In this work, we study the continuity of pseudodifferential operators on local Hardy spaces h p (ℝ n ) and generalize the results due to Goldberg and Taylor by showing that operators with symbols in S 1,δ 0(ℝ n ), 0≤δ<1, and in some subclasses of S 1,10(ℝ n ) are bounded on h p (ℝ n ) (0<p≤1). As an application, we study the local solvability of the planar vector field L= t +ib(x,t) x , b(x,t)≥0, in spaces of mixed norm involving Hardy spaces. Work supported in part by CNPq, FINEP, and FAPESP.  相似文献   

15.
In this paper the Cauchy problem for a class of nonhomogeneous Navier-Stokes equations in the infinite cylinderS T =ℝn x [0,T) is considered. We construct a unique local solution inL q([0,T);L p (ℝ n )) for a class of nonhomogeneous Navier-Stokes equations provided that initial data are inL r (ℝ n ), wherer>1 is an exponent determined by the structure of nonlinear terms andp,q are such that 2/q=n(1/r−1/p). Meanwhile under suitable conditions we also obtain thatu(t)L q([0,∞];L p (ℝ n )) provided that initial data are sufficiently small. This work is supported by the National Natural Sciences Foundation of China and the Foundation of LNM Laboratory of Institute of Mechanics of the Chinese Academy of Sciences.  相似文献   

16.
In this paper, it was proved that the commutator generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L p1 (ℝ n ) to L p2 (ℝ n ) if and only if b is a CṀO(ℝ n ) function, where 1/p 1 − 1/p 2 = β/n, 1 < p 1 < ∞, 0 ⩽ β < n. Furthermore, the characterization of on the homogenous Herz space (ℝ n ) was obtained. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10571014, 10371080) and the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No. 20040027001)  相似文献   

17.
Let A be a symmetric expansive matrix and Hp(Rn) be the anisotropic Hardy space associated with A. For a function m in L∞(Rn), an appropriately chosen function η in Cc∞(Rn) and j ∈ Z define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 < p < 1 and (m)j belongs to the anisotropic nonhomogeneous Herz space K11/p-1,p(Rn), then m is a Fourier multiplier from Hp(Rn) to Lp(Rn). For p = 1, a similar result is obtained if the space K10,1(Rn) is replaced by a slightly smaller space K(w).Moreover, the authors show that if 0 < p ≤ 1 and if the sequence {(mj)V} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp(Rn) to Lp(Rn).  相似文献   

18.
Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn) with suitable upper bounds on its heat kernels. Assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper,we define the Littlewood- Paley g function associated with L on Rn × Rn, denoted by GL(f)(x1, x2), and decomposition, we prove that ‖SL(f)‖p ≈‖GL(f)‖p ≈‖f‖p for 1 < p <∞.  相似文献   

19.
LetH n ≅ℝ2n ⋉ℝ be the Heisenberg group and letμ t be the normalized surface measure for the sphere of radiust in ℝ2n . Consider the maximal function defined byM f=sup t>0|f*μ t |. We prove forn≥2 thatM defines an operator bounded onL p (H n ) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL p boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type. The second author was supported in part by the National Science Foundation.  相似文献   

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