Data In Besov Spaces Research Articles

On the global well-posedness for the compressible Hall-MHD system

In this paper, we prove the global well-posedness of the Cauchy problem to the 3-D compressible Hall-magnetohydrodynamic (MHD) system supplemented with a class of large initial data in Besov spaces. By exploiting the intrinsic structure of the compressible Hall-MHD equations, we obtain the diffusion and the dispersive effects for the mixed hyperbolic-parabolic system. Based on this important observation, we prove that the compressible Hall-MHD system exists a global solution with the initial data close to a stable equilibrium.

Non-uniform Continuity of the Generalized Camassa–Holm Equation in Besov Spaces

In the paper, we gave a strengthening of our previous work in \cite{Li1} (J. Differ. Equ. 269 (2020)) and proved that the data-to-solution map for the Camassa-Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\R)$ with $s>\max\{1+1/{p},3/2\}$ and $(p,r)\in [1,\infty]\times[1,\infty)$. The method applies also to the b-family of equations which contain the Camassa-Holm and Degasperis-Procesi equations.

Non-uniform continuity on initial data for the two-component b-family system in Besov space

This paper studies a two-component b-family system, which includes the two-component Camassa-Holm system and the two-component Degasperis-Procesi system as special case. It is shown that the solution map of this system is not uniformly continuous on the initial data in Besov spaces \(B_{p, r}^{s-1}({\mathbb {R}})\times B_{p, r}^s({\mathbb {R}})\) with \(s>\max \{1+\frac{1}{p}, \frac{3}{2}\}\), \(1\le p, r< \infty \). Our result covers and extends the previous non-uniform continuity in Sobolev spaces \(H^{s-1}({\mathbb {R}})\times H^s({\mathbb {R}})\) for \(s>\frac{5}{2}\) to Besov spaces (Nonlinear Anal., 2014, 111: 1-14). Compared with the generalized rotation b-family system considered by Holmes et al. (Z. Angew. Math. Mech., 2021), our non-uniform continuity is established in a broader range of Besov spaces.

A note on the initial value problem for a higher‐order Camassa–Holm equation

AbstractConsidered herein is the Cauchy problem for a higher‐order Camassa–Holm equation. Based on the local well‐posedness results for this problem, the non‐uniformly continuous dependence on initial data is established in Sobolev spaces with on the line by using the method of approximate solutions. In the periodic case, the non‐uniformly continuous dependence on initial data in Besov spaces and are proved. Finally, the persistence property of solutions for this problem is studied.

Gevrey Regularity and Time Decay of Fractional Porous Medium Equation in Critical Besov Spaces

In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 α ≤ 2, we prove global well-posedness for initial data with 1 ≤ p q ≤ ∞, and analyticity of solutions with 1 p q ≤ ∞. In particular, we also proved that when α = 1, both u and belong to . We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is as t → ∞.

Non-uniform continuity of the Fokas–Olver–Rosenau–Qiao equation in Besov spaces

In this paper, we consider the solution map of the Cauchy problem to the Fokas–Olver–Rosenau–Qiao equation on the real line and prove that the solution map of this problem is not uniformly continuous on the initial data in Besov spaces. Our result extends the previous results in Himonas and Mantzavinos (Nonlinear Anal 95:499–529, 2014) and Li et al. (J Math Fluid Mech 22:50, 2020).

Non-uniform Dependence on Initial Data for the Generalized Camassa–Holm–Novikov Equation in Besov Space

Considered in this paper is the generalized Camassa-Holm-Novikov equation with high order nonlinearity, which unifies the Camassa-Holm and Novikov equations as special cases. We show that the solution map of generalized Camassa-Holm-Novikov equation is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^s(\mathbb{R})$ with $s>\max\{1+\frac{1}{p}, \frac{3}{2}\}$, $1\leq p, r< \infty$ as well as in critical space $B_{2, 1}^{\frac{3}{2}}(\mathbb{R}).$ Our result covers and improves the previous work given by Li et al. \cite{Li 2020, 1Li 2020, Li 2021}(J. Differ. Equ. 269 (2020) 8686-8700; J. Math. Fluid Mech. 22 (2020) 4:50; J. Math. Fluid Mech., (2021) 23:36).

Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

Abstract We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein–Gordon and fractional Schrödinger equations. Our estimates extend those of Frank–Sabin in the case of the wave and Klein–Gordon equations, and generalize work of Frank et al. and Frank–Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results. The main novelty of this paper is our derivation and use of estimates for weighted oscillatory integrals, which we combine with an approach due to Frank and Sabin. Our weighted oscillatory integral estimates are, in a certain sense, rather delicate endpoint versions of known dispersive estimates with power-type weights of the form $|\xi |^{-\lambda }$ or $(1 + |\xi |^2)^{-\lambda /2}$ , where $\lambda \in \mathbb {R}$ . We achieve optimal decay rates by considering such weights with appropriate $\lambda \in \mathbb {C}$ . For the wave and Klein–Gordon equations, our weighted oscillatory integral estimates are new. For the fractional Schrödinger equation, our results overlap with prior work of Kenig–Ponce–Vega in a certain regime. Our contribution to the theory of weighted oscillatory integrals has also been influenced by earlier work of Carbery–Ziesler, Cowling et al., and Sogge–Stein. Finally, we provide some applications of our new Strichartz estimates for orthonormal families of data to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces.

Non-uniform Dependence for the Novikov Equation in Besov Spaces

In this paper, we investigate the dependence on initial data of solutions to the Novikov equation. We show that the solution map is not uniformly continuous dependence on the initial data in Besov spaces $$B^s_{p,r}({\mathbb {R}}), s>\max \{1+\frac{1}{p},\frac{3}{2}\}$$ .

Small data global regularity for half-wave maps

We formulate the half-wave maps problem with target $S^2$ and prove global regularity in sufficiently high spatial dimensions for a class of small critical data in Besov spaces.

Interpolation and Embeddings of Weighted Tent Spaces

Given a metric measure space X, we consider a scale of function spaces T^{p,q}_s(X), called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on X we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call Z-spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy–Littlewood–Sobolev-type embeddings between weighted tent spaces.

Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted L classes. We establish: (1) Mapping properties for the double and single layer potentials, as well as the Newton potential; (2) Extrapolation-type solvability results: the fact that solvability of the Dirichlet or Neumann boundary value problem at any given L space automatically assures their solvability in an extended range of Besov spaces; (3) Well-posedness for the non-homogeneous boundary value problems. In particular, we prove well-posedness of the non-homogeneous Dirichlet problem with data in Besov spaces for operators with real, not necessarily symmetric coefficients. 2010 Mathematics Subject Classification. Primary 35J25, Secondary 31B20, 35C15, 46E35.

Local Well-Posedness of a Coupled Camassa-Holm System in Critical Spaces

In this paper, we consider the local well-posedness of a coupled Camassa-Holm system with initial data in Besov spaces B^s_{2,1} with critical index s=\frac{3}{2}.

ON THE WELL-POSEDNESS OF THE FULL LOW MACH NUMBER LIMIT SYSTEM IN GENERAL CRITICAL BESOV SPACES

This work is devoted to the well-posedness issue for the low Mach number limit system obtained from the full compressible Navier–Stokes system, in the whole space ℝd with d ≥ 2. In the case where the initial temperature (or density) is close to a positive constant, we establish the local existence and uniqueness of a solution in critical homogeneous Besov spaces of type [Formula: see text]. If, in addition, the initial velocity is small then we show that the solution exists for all positive time. In the fully nonhomogeneous case, we establish the local well-posedness in nonhomogeneous Besov spaces [Formula: see text] (still with critical regularity) for arbitrarily large data with positive initial temperature. Our analysis strongly relies on the use of a modified divergence-free velocity which allows to reduce the system to a nonlinear coupling between a parabolic equation and some evolutionary Stokes system. As in the recent work by Abidi and Paicu [Existence globale pour un fluide inhomogène, Ann. Inst. Fourier 57(3) (2007) 883–917]. Concerning the density-dependent incompressible Navier–Stokes equations, the Lebesgue exponents of the Besov spaces for the temperature and the (modified) velocity, need not be the same. This enables us to consider initial data in Besov spaces with a negative index of regularity.

The dirichlet problem in lipschitz domains for higher order elliptic systems with rough coefficients

We study the Dirichlet problem, in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order with bounded, complex-valued coefficients. A sharp corollary of our main solvability result is that the operator of this problem performs an isomorphism between weighted Sobolev spaces when its coefficients and the unit normal of the boundary belong to the space VMO.

Well-posedness in critical spaces for incompressible viscoelastic fluid system

In this paper, we study the well-posedness in critical spaces of incompressible viscoelastic fluid system of Oldroyd model. Precisely, for this system with initial data in Besov space of critical regularity, we prove the existence and uniqueness of the local solution, which is also shown to exist globally in time provided the initial data is small under certain norms.

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&lt;∞ 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&gt;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&lt;p&lt;∞ and 1⩽r&lt;∞. Copyright © 2008 John Wiley &amp; Sons, Ltd.

On the Cauchy problem of fourth-order nonlinear Schrödinger equations

In this paper we study the Cauchy problem of the fourth-order Schrödinger equation i ∂ t u + a Δ u + b Δ 2 u = ± u p for dimension ≤ 4 , where p is an integer greater than 1, with initial data in Besov spaces. We prove that for any 4 ( p 2 − 1 ) ( 4 − n ) p + 4 + n ≤ q ≤ ∞ , the Cauchy problem of this equation is locally well-posed in B ̇ 2 , q s p ( R n ) and B 2 , q s ( R n ) , where s p = n 2 − 4 p − 1 and s > s p , and for any 1 ≤ q ≤ ∞ almost global well-posedness holds in these spaces if initial data are small. We also prove that if a = 0 , then global well-posedness holds in these spaces for small initial data.

Layer potentials and boundary value problems for Laplacian in Lipschitz domains with data in quasi-Banach Besov spaces

We study the Dirichlet and Neumann boundary value problems for the Laplacian in a Lipschitz domain \({\Omega}\), with boundary data in the Besov space \({B_{s}^{p,p} (\partial\Omega).}\) The novelty is to identify a way of measuring smoothness for the solution u that allows us to consider the case p 1 was treated.

Solvability of the Cauchy problem of nonlinear beam equations in Besov spaces

In this paper we study solvability of the Cauchy problem of the nonlinear beam equation ∂ t 2 u + △ 2 u = ± u p with initial data in Besov spaces. We prove that, for any 1 ≤ q < ∞ , the Cauchy problem of this equation is locally well-posed in the Besov spaces B ̇ 2 , q s p ( R n ) and B 2 , q s ( R n ) , where s p = n 2 − 4 p − 1 and s > s p , and globally well-posed in these spaces if initial data are small. Moreover we obtain scattering results in B ̇ 2 , q s p ( R n ) .