Framework Of Besov Spaces Research Articles

The optimal decay estimates on the framework of Besov spaces for the Euler–Poisson two-fluid system

In this paper, we are concerned with the optimal decay estimates for the Euler–Poisson two-fluid system. It is first revealed that the irrotationality of the coupled electronic field plays a key role such that the two-fluid system has the same dissipative structure as generally hyperbolic systems satisfying the Shizuta–Kawashima condition. This fact inspires us to obtain decay properties for linearized systems in the framework of Besov spaces. Furthermore, various decay estimates of solution and its derivatives of fractional order are deduced by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As the direct consequence, the optimal decay rates of Lp(ℝ3)-L2 (ℝ3) (1 ≤ p < 2) type for the Euler–Poisson two-fluid system are also shown. Compared with previous works in Sobolev spaces, a new observation is that the difference of variables exactly consists of a one-fluid Euler–Poisson equations, which leads to the sharp decay estimates for velocities.

Global existence and optimal decay rates for the Timoshenko system: The case of equal wave speeds

We first show the global existence and optimal decay rates of solutions to the classical Timoshenko system in the framework of Besov spaces. Due to the non-symmetric dissipation, the general theory for dissipative hyperbolic systems (see [31]) cannot be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequently, the optimal decay estimates of solution and its derivatives of fractional order are shown by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As a by-product, the usual decay estimate of L1(R)-L2(R) type is also shown.

Well-posedness and time-decay for compressible viscoelastic fluids in critical Besov space

In this paper, we construct local solution with highly oscillating initial velocity and then get the global strong solution in the Lp based Besov space which improves the result of J. Qian, Z. Zhang (2010) [25] and X. Hu, D. Wang (2011) [14]. The local existence and uniqueness lies on the Lagrange coordinate transform and the contraction mapping theorem. The global result lies on a decomposition of the system and some commutator estimates. In the last part, we prove a time-decay in the critical Besov space framework which seems to have little investigation. The proof is based on the properties of the Green's matrix and various interpolations between Besov type spaces.

Boundary values of the resolvent of Schrödinger hamiltonians with potentials of order zero

Let $H=-\Delta +V$ be a Schrödinger hamiltonian acting on $L^2(\mathbb{R}^n)$, $n\geq 2$, where $V$ a potential of order zero plus a short-range perturbation. In this work we investigate the behavior of the resolvent $R(z)=(H-z)^{-1}$ of $H$ as Im$\,z \downarrow 0$, at high energies and in the framework of Besov spaces $B(\mathbb{R}^n)$.For $\lambda_0>0$ sufficiently large and $\lambda\geq\lambda_0$, we show that there exists a linear operator $R(\lambda+i0)$ such that $R(\lambda+i\epsilon)$ converges to $R(\lambda+i0)$ as $\epsilon\downarrow 0$, strongly in $\mathcal{L}(L^{2, s}(\mathbb{R}^n),L^{2,-s}(\mathbb{R}^n))$, $s>1/2$, and weakly in $\mathcal{L}(B(\mathbb{R}^n),B^*(\mathbb{R}^n))$. We achieve this through a Mourre-estimate strategy.

Well-Posedness for a Multidimensional Viscous Liquid-Gas Two-Phase Flow Model

The Cauchy problem of a multidimensional ($d\geqslant 2$) compressible viscous liquid-gas two-phase flow model is studied in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local-in-time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.

Strong relaxation limit of multi-dimensional isentropic Euler equations

This paper is devoted to study the strong relaxation limit of multi-dimensional isentropic Euler equations with relaxation. Motivated by the Maxwell iteration, we generalize the analysis of Yong (SIAM J Appl Math 64:1737–1748, 2004) and show that, as the relaxation time tends to zero, the density of a certain scaled isentropic Euler equations with relaxation strongly converges towards the smooth solution to the porous medium equation in the framework of Besov spaces with relatively lower regularity. The main analysis tool used is the Littlewood–Paley decomposition.

On Dilation Operators in Besov Spaces

We consider dilation operators T k : f → f(2 k .) in the framework of Besov spaces B s p,q (R n ) when 0 n(1/p — 1) Tk is a bounded linear operator from B s p,q (R n ) into itself and there are optimal bounds for its norm. We study the situation on the line s = n(1/p — 1), an open problem mentioned in [5, 2.3.1, 2.3.2]. It turns out that the results shed new light upon the diversity of different approaches to Besov spaces on this line, associated to definitions by differences, Fourier-analytical methods, and subatomic decompositions.

Global existence for the magnetohydrodynamic system in critical spaces

In this article, we show that the magnetohydrodynamic system in . We also prove that the constructed solution exists globally in time if the initial data are small. In particular, this allows us to work in the framework of Besov space with negative regularity indices and this fact is particularly important when the initial data are strongly oscillating.

On the well‐posedness of the Cauchy problem for an MHD system in Besov spaces

AbstractThis paper is devoted to the study of the Cauchy problem of incompressible magneto‐hydrodynamics system in the framework of Besov spaces. In the case of spatial dimension n⩾3, we establish the global well‐posedness of the Cauchy problem of an incompressible magneto‐hydrodynamics system for small data and the local one for large data in the Besov space Ḃ (ℝn), 1⩽p<∞ and 1⩽r⩽∞. Meanwhile, we also prove the weak–strong uniqueness of solutions with data in Ḃ (ℝn)∩L2(ℝn) for n/2p+2/r>1. In the case of n=2, we establish the global well‐posedness of solutions for large initial data in homogeneous Besov space Ḃ (ℝ2) for 2<p<∞ and 1⩽r<∞. Copyright © 2008 John Wiley & Sons, Ltd.

Existence and asymptotic behavior of C1 solutions to the multi-dimensional compressible Euler equations with damping

In this paper, the existence and asymptotic behavior of C 1 solutions to the multi-dimensional compressible Euler equations with damping on the framework of Besov space are considered. Comparing with the well-posedness results of Sideris–Thomases–Wang [T. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the three-dimensional compressible Euler with damping, Comm. Partial Differential Equations 28 (2003) 953–978], we weaken the regularity assumptions on the initial data. The global existence lies on a crucial a-priori estimate which is obtained by the spectral localization method. The main analytic tools are the Littlewood–Paley decomposition and Bony’s paraproduct formula.

GLOBAL EXPONENTIAL STABILITY OF CLASSICAL SOLUTIONS TO THE HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

In this paper, the global well-posedness and stability of classical solutions to the multidimensional hydrodynamic model for semiconductors on the framework of Besov space are considered. We weaken the regularity requirement of the initial data, and improve some known results in Sobolev space. The local existence of classical solutions to the Cauchy problem is obtained by the regularized means and compactness argument. Using the high- and low-frequency decomposition method, we prove the global exponential stability of classical solutions (close to equilibrium). Furthermore, it is also shown that the vorticity decays to zero exponentially in the 2D and 3D space. The main analytic tools are the Littlewood–Paley decomposition and Bony's para-product formula.

Modeling Occlusion and Scaling in Natural Images

The dead leaves model, introduced by the mathematical morphology school, consists of the superposition of random closed sets (the objects) and enables one to model the occlusion phenomena. When combined with specific size distributions for objects, one obtains random fields providing adequate models for natural images. However, this framework imposes bounds on the sizes of objects. We consider the limits of these random fields when letting the cutoff sizes tend to zero and infinity. As a result we obtain a random field that contains homogeneous regions, satisfies scaling properties, and is statistically relevant for modeling natural images. We then investigate the combined effect of these features on the regularity of images in the framework of Besov spaces.

Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations

In this paper the global well-posedness in $L^2$ and $H^m$ of the Cauchy problem is proved for nonlinear Schrodinger-type equations. This we do by establishing regular Strichartz estimates for the corresponding linear equations and some nonlinear a priori estimates in the framework of Besov spaces. We further establish the regularity of the $H^m$-solution to the Cauchy problem.

Hs-global well-posedness for semilinear wave equations

We consider the Cauchy problem for semilinear wave equations in R n with n⩾3. Making use of Bourgain's method in conjunction with the endpoint Strichartz estimates of Keel and Tao, we establish the H s -global well-posedness with s<1 of the Cauchy problem for the semilinear wave equation. In doing so a number of nonlinear a priori estimates is established in the framework of Besov spaces. Our method can be easily applied to the case with n=3 to recover the result of Kenig–Ponce–Vega.

Commutator estimates, Besov spaces and scattering problems for the acoustic wave propagation in perturbed stratified fluids

Mourre's commutator method is used together with Besov spaces, in this paper, to settle the question of uniqueness of solutions to the scattering problem for the acoustic wave propagation in perturbed stratified fluids in Rn with n [ges ] 2: The uniqueness of solutions is established by introducing a radiation condition in the framework of Besov spaces, and is then used to prove resolvent estimates in the Besov space setting for the acoustic propagator in perturbed stratified fluids. These estimates are ‘sharp’ in a sense made precise in the text and are important in establishing existence of solutions to the scattering problem.