Littlewood-Paley Decomposition Research Articles

On the well-posedness of the quasi-geostrophic equation in the Triebel-Lizorkin-Lorentz spaces

In this paper, we establish the local well-posedness for the quasi-geostrophic equations and obtain a blow-up criterion of smooth solutions in the framework of Triebel-Lizorkin-Lorentz spaces by adapting a method in Chen-Miao-Zhang (Arch. Rational Mech. Anal. 195: 2010, 561–578). Our new function spaces contain the classical Triebel-Lizorkin spaces and Sobolev spaces, and thus the corresponding results generalize several known ones, for instance, Chae (Nonlinearity 16: 2003, 479–495) and Castro et al. (Nonlinearity 22: 2009, 1791–1815). The main ingredients of our proofs are Littlewood–Paley decomposition and the paradifferential calculus.

Universal Bounds for the Littlewood-Paley First-Order Moments of the 3D Navier-Stokes Equations

We derive upper bounds for the infinite-time and space average of the L1-norm of the Littlewood-Paley decomposition of weak solutions of the 3D periodic Navier-Stokes equations. The result suggests that the Kolmogorov characteristic velocity scaling, \({\mathbf{U}_\kappa\sim\epsilon^{1/3} \kappa^{-1/3}}\) , holds as an upper bound for a region of wavenumbers near the dissipative cutoff.

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.

Remarks on Littlewood–Paley Analysis

AbstractLittlewood–Paley analysis is generalized in this article. We show that the compactness of the Fourier support imposed on the analyzing function can be removed. We also prove that the Littlewood–Paley decomposition of tempered distributions converges under a topology stronger than the weak-star topology, namely, the inductive limit topology. Finally, we construct a multiparameter Littlewood–Paley analysis and obtain the corresponding “renormalization” for the convergence of this multiparameter Littlewood–Paley analysis.

On the classical solutions to the hydrodynamic model for semiconductors

This paper mainly deals with the multidimensional hydrodynamic model for semiconductors. Inspired by the previous papers [Y. Shizuta, S. Kawashima, Systems of equations of hyperbolic–parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249–275; S. Kawashima, W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal. 174 (2004) 345–364; W.-A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal. 172 (2004) 247–266], we develop some new frequency-localization estimates to establish the global existence and exponential stability of (small) classical solutions in a class of critical Besov spaces, which are different from estimates in our recent paper [D.Y. Fang, J. Xu, T. Zhang, Global exponential stability of classical solutions to the hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci. 17 (2007) 1507–1530]. Furthermore, this new method can also be applied to the multidimensional Euler equations with damping. The analytic tool used is the Littlewood–Paley decomposition.

Energy conservation and Onsager's conjecture for the Euler equations

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in conserve energy only if they have a certain minimal smoothness (of the order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space . We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood–Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.

A weighted L q -approach to Stokes flow around a rotating body

Considering time-periodic Stokes flow around a rotating body in $${\mathbb R^2}$$ or $${\mathbb R^3}$$ we prove weighted a priori estimates in L q -spaces for the whole space problem. After a time-dependent change of coordinates the problem is reduced to a stationary Stokes equation with the additional term $${(\omega \times x)\cdot\nabla u}$$ in the equation of momentum, where ω denotes the angular velocity. In cylindrical coordinates attached to the rotating body we allow for Muckenhoupt weights which may be anisotropic or even depend on the angular variable and prove weighted L q -estimates using the weighted theory of Littlewood-Paley decomposition and of maximal operators.

Existence theorem and blow‐up criterion of the strong solutions to the magneto‐micropolar fluid equations

AbstractIn this paper we study the magneto‐micropolar fluid equations in ℝ3, prove the existence of the strong solution with initial data inHs(ℝ3) for$s>{3\over2}$, and set up its blow‐up criterion. The tool we mainly use is Littlewood–Paley decomposition, by which we obtain a Beale–Kato–Majda‐type blow‐up criterion for smooth solution (u, ω,b) that relies on the vorticity of velocity ∇ × uonly. Copyright © 2007 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.

Strichartz Inequalities for the Wave Equation with the Full Laplacian on the Heisenberg Group

AbstractWe prove dispersive and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on the Heisenberg group, by means of Besov spaces defined by a Littlewood–Paley decomposition related to the spectral resolution of the full Laplacian. This requires a careful analysis due also to the non-homogeneous nature of the full Laplacian. This result has to be compared to a previous one by Bahouri, Gérard and Xu concerning the solution of the wave equation related to the Kohn Laplacian.

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.

The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations

We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, namely $$\sup_{j\in\mathbb{Z}}\int_0^T\|\Delta_j(\nabla\times u)\|_\infty dt,$$where Δ j is the frequency localization operator in the Littlewood-Paley decomposition.

A weighted Lq‐approach to Oseen flow around a rotating body

AbstractWe study time‐periodic Oseen flows past a rotating body in ℝ3 proving weighted a priori estimates in Lq‐spaces using Muckenhoupt weights. After a time‐dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional terms (ω ∧ x) ⋅ ∇ u and −ω ∧ u in the equation of momentum where ω denotes the angular velocity. Due to the asymmetry of Oseen flow and to describe its wake we use anisotropic Muckenhoupt weights, a weighted theory of Littlewood–Paley decomposition and of maximal operators as well as one‐sided univariate weights, one‐sided maximal operators and a new version of Jones' factorization theorem. Copyright © 2007 John Wiley & Sons, Ltd.

A Note on the Relaxation-Time Limit of the Isothermal Euler Equations

This work is concerned with the relaxation-time limit of the multidimensional isothermal Euler equations with relaxation. We show that Coulombel-Goudon's results (2007) can hold in the weaker and more general Sobolev space of fractional order. The method of proof used is the Littlewood-Paley decomposition.

SMOOTHNESS OF WEAK SOLUTIONS OF THE SPATIALLY HOMOGENEOUS LANDAU EQUATION

We prove that any weak solution (in the sense of Desvillettes and Villani [6]) to the Landau homogeneous equation is smooth and, in particular, that it lies in Schwartz space [Formula: see text]. Our result has to be compared to [6], where Desvillettes and Villani have shown existence of smooth weak solutions along with a uniqueness result, under further assumptions on initial data. Our method of proof uses the Littlewood–Paley decomposition of velocity space, and elementary Cauchy–Schwarz or Young's inequalities, thus avoiding any use of complex Sobolev interpolation inequalities.

A dyadic decomposition approach to a finitely degenerate hyperbolic problem

We use the Littlewood-Paley decomposition technique to obtain a C∞-well-posedness result for a weakly hyperbolic equation with a finite order of degeneration. Keywords: Littlewood-Paley decomposition, Hyperbolic equations, C∞-well-posedness, Approximate energy method

Littlewood–Paley decompositions and Besov spaces on Lie groups of polynomial growth

AbstractWe introduce a Littlewood–Paley decomposition related to any sub‐Laplacian on a Lie group G of polynomial volume growth; this allows us to prove a Littlewood–Paley theorem in this general setting and to provide a dyadic characterization of Besov spaces B s,q p (G ), s ∈ ℝ, equivalent to the classical definition through the heat kernel. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Space-Time Estimates in the Besov Spaces and the Navier-Stokes Equations

In this paper, we establish the space-time estimates in the Besov spaces of the solution to the Navier-Stokes equations in $\bold R^n , n\geq 3$. As an application, we improve some known results about the regularity criterion of weak solutions and the blow-up criterion of smooth solutions. Our main tools are the frequency localization and the Littlewood-Paley decomposition.

On the system of conservation laws and its perturbation in the Besov spaces

We prove that a system of conservation laws on $ \mathbb R^N$ is locally well posed in the `critical' Besov space $B^{\frac{N}{2} +1}_{2,1}$. The time of local existence depends only on the size of the inhomogeneous part of the initial data. We also obtain a blow-up criterion of the local solution. For the conservation system with a dissipation term added we prove global existence of solutions under the assumption of smallness of the homogeneous part of the arbitrary-sized $B^{\frac{N}{2} -1}_{2,1}$ norm of the initial data. For the proof of these results we essentially use the Littlewood-Paley decomposition of functions to derive the energy type of estimate.

On the Euler Equations in the Critical Triebel-Lizorkin Spaces

In this paper we study the initial value problem of the incompressible Euler equations in ℝ n for initial data belonging to the critical Triebel-Lizorkin spaces, i.e., v 0 F n+1 1,q , q[1, ∞]. We prove the blow-up criterion of solutions in F n+1 1,q for n=2,3. For n=2, in particular, we prove global well-posedness of the Euler equations in F 3 1,q , q[1, ∞]. For the proof of these results we establish a sharp Moser-type inequality as well as a commutator-type estimate in these spaces. The key methods are the Littlewood-Paley decomposition and the paradifferential calculus by J. M. Bony.