Besov Spaces Research Articles

Stability of stationary solutions to the Navier–Stokes equations in the Besov space

AbstractWe consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space for . It is clarified that if w is small in for and , then for every small initial disturbance with and (), there exists a unique solution of the nonstationary Navier–Stokes equations on (0, ∞) with such that and as , for , , and small .

Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embeddings between the Besov spaces defined by differences and by Fourier-analytical decompositions as well as between Besov and Sobolev/Triebel-Lizorkin spaces; Various new characterizations for Besov norms in terms of different K-functionals. For instance, we derive characterizations via ball averages, approximation methods, heat kernels, and Bianchini-type norms; Sharp estimates for Besov norms of derivatives and potential operators (Riesz and Bessel potentials) in terms of norms of functions themselves. We also obtain quantitative estimates of regularity properties of the fractional Laplacian. The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

Local existence for the d‐dimensional magneto‐micropolar equations with fractional dissipation in Besov spaces

In this paper, we consider the Cauchy problem of the d‐dimensional magneto‐micropolar equations ( or ) with general fractional dissipation. The aim of this paper is to obtain the existence and uniqueness of solutions in the weakest possible inhomogeneous Besov spaces. Using the technical tools of Litttlewood–Paley decomposition and Besov spaces theory, we obtain the local existence in the functional setting of inhomogeneous Besov spaces. Furthermore, such solutions are unique only in 2D case.

Stationary solutions for the fractional Navier–Stokes–Coriolis system in Fourier–Besov spaces

In this work we prove the existence of stationary solutions for the tridimensional fractional Navier–Stokes–Coriolis in critical Fourier–Besov spaces. We first deal with the non-stationary fractional Navier–Stokes–Coriolis and in this framework we get the existence of stationary solutions. Also we state a kind of stability of these non-stationary solutions which applied to the stationary case permits to conclude that, under suitable conditions, non-stationary solutions converge to the stationary ones when the time goes to infinity. Finally we establish a relation between the external force and the Coriolis parameter in order to get a unique solution for the stationary system.

Generalized Jacobian Weighted Modulus of Smoothness

The topic of modulus of smoothness still gets the interest of many researchers due to its applicable usage in different fields, especially for function approximation. In this paper, we define a new modulus of smoothness of weighted type. The properties of our modulus are studied. These properties can be easily used in different fields, in particular, the functions in the Besov spaces when

A uniqueness property for Bergman functions on the Siegel upper half-space

In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if f ∈ A t p ( U ) f\in A_t^p(\mathcal {U}) and L α f ≡ 0 \mathcal {L}^{\alpha } f\equiv 0 for some nonnegative multi-index α \alpha , then f ≡ 0 f\equiv 0 , where L α ≔ ( L 1 ) α 1 ⋯ ( L n ) α n \mathcal {L}^{\alpha }≔(\mathcal {L}_1)^{\alpha _1} \cdots (\mathcal {L}_n)^{\alpha _n} with L j = ∂ ∂ z j + 2 i z ¯ j ∂ ∂ z n \mathcal {L}_j = \frac {\partial }{\partial z_j} + 2i \bar {z}_j \frac {\partial }{\partial z_n} for j = 1 , … , n − 1 j=1,\ldots , n-1 and L n = ∂ ∂ z n \mathcal {L}_n = \frac {\partial }{\partial z_n} . As a consequence, we obtain a new integral representation for the Bergman functions on the Siegel upper half-space. In the end, as an application, we derive a result that relates the Bergman norm to a “derivative norm”, which suggests an alternative definition of the Bloch space and a notion of the Besov spaces over the Siegel upper half-space.

Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth

Abstract We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form min ⁡ { ∫ Ω F ⁢ ( x , D ⁢ w ) ⁢ 𝑑 x : w ∈ 𝒦 ψ ⁢ ( Ω ) } , \min\bigg{\{}\int_{\Omega}F(x,Dw)dx:w\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where Ω is a bounded open set of ℝ n {\mathbb{R}^{n}} , n ≥ 2 {n\geq 2} , the function ψ ∈ W 1 , p ⁢ ( Ω ) {\psi\in W^{1,p}(\Omega)} is a fixed function called obstacle, and 𝒦 ψ ⁢ ( Ω ) := { w ∈ W 1 , p ⁢ ( Ω ) : w ≥ ψ ⁢ a.e. in ⁢ Ω } \mathcal{K}_{\psi}(\Omega):=\bigl{\{}w\in W^{1,p}(\Omega):w\geq\psi\text{ a.e.% in }\Omega\bigr{\}} is the class of admissible functions. If the obstacle ψ is locally bounded, we prove that the gradient of solution inherits some fractional differentiability property, assuming that both the gradient of the obstacle and the mapping x ↦ D ξ ⁢ F ⁢ ( x , ξ ) {x\mapsto D_{\xi}F(x,\xi)} belong to some suitable Besov space. The main novelty is that the Besov type regularity on the partial map x ↦ D ξ ⁢ F ⁢ ( x , ξ ) {x\mapsto D_{\xi}F(x,\xi)} is not related to the dimension n.

Embeddings of anisotropic Besov spaces into Sobolev spaces

AbstractWe study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix A into Sobolev spaces, with a focus on the influence of A on the embedding behavior. For a large range of parameters, we derive sharp characterizations of embeddings.

A Derivative-Free Characterization of the Weighted Besov Spaces

A Derivative-Free Characterization of the Weighted Besov Spaces

The two‐dimensional stationary Navier–Stokes equations in toroidal Besov spaces

AbstractWe consider the stationary Navier–Stokes equations in the two‐dimensional torus . For any , we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces for given small external forces in when . These spaces become closer to the scaling invariant ones if the difference ε becomes smaller. This well‐posedness is proved by using the embedding property and the para‐product estimate in homogeneous Besov spaces. In addition, for the case , we can show the ill‐posedness, even in the scaling invariant spaces. Actually in such cases of p and q, we can prove that ill‐posedness by showing the discontinuity of a certain solution map from to .

A remark on the regularity criterion for the 3D Navier–Stokes equations in terms of two vorticity components

In this paper, we establish a new regularity criterion for the 3D Navier–Stokes equations in terms of only two vorticity components. By making use of a logarithmic Sobolev inequality in the Besov spaces with negative indices and well known commutator estimate, we prove that a unique local strong solution does not blow-up at time T if two components of vorticity belong to L2(0,T;Ḃ∞,∞−1). This result is the further extension of the previous works by Guo et al. (2018) and by O (2021), and becomes the improvement of the work by Gala (2011).

Continuous characterization of the Besov spaces of variable smoothness and integrability

UDC 517.9 We obtain new equivalent quasinorms of the Besov spaces of variable smoothness and integrability. Our main tools are the continuous version of the Calderón reproducing formula, maximal inequalities, and the variable-exponent technique; however, allowing the parameters to vary from point to point leads to additional difficulties which, in general, can be overcome by imposing regularity assumptions on these exponents.

Boundary conjugation problem for piecewise analytic functions in Besov spaces

The main aim of this work is to show the solvability of the homogeneous and inhomogeneous boundary conjugation problems for piecewise analytic functions in the Besov space, which is embedded into the class of continuous functions.

Noncommutative [formula omitted] functions and Fréchet derivatives of operator functions

Fix a unital C∗-algebra A, and write Asa for the set of self-adjoint elements of A. Also, if f:R→ℂ is a continuous function, then write fA:Asa→A for the operator functiona↦f(a) defined via functional calculus. In this paper, we introduce and study a space NCk(R) of Ck functions f:R→ℂ such that, no matter the choice of A, the operator function fA:Asa→A is k-times continuously Fréchet differentiable. In other words, if f∈NCk(R), then f “lifts” to a Ck map fA:Asa→A, for any (possibly noncommutative) unital C∗-algebra A. For this reason, we call NCk(R) the space of noncommutativeCkfunctions. Our proof that fA∈Ck(Asa;A), which requires only knowledge of the Fréchet derivatives of polynomials and operator norm estimates for “multiple operator integrals” (MOIs), is more elementary than the standard approach; nevertheless, NCk(R) contains all functions for which comparable results are known. Specifically, we prove that NCk(R) contains the homogeneous Besov space Ḃ1k,∞(R) and the Hölder space Clock,ɛ(R). We highlight, however, that the results in this paper are the first of their type to be proven for arbitrary unital C∗-algebras, and that the extension to such a general setting makes use of the author’s recent resolution of certain “separability issues” with the definition of MOIs. Finally, we prove by exhibiting specific examples that Wk(R)loc⊊NCk(R)⊊Ck(R), where Wk(R)loc is the “localized” kth Wiener space.

Function spaces via fractional Poisson kernel on Carnot groups and applications

We provide a new characterization of homogeneous Besov and Sobolev spaces in Carnot groups using the fractional heat kernel and Poisson kernel. We apply our results to study commutators involving fractional powers of the sub-Laplacian.

Nonlinear wave equations with slowly decaying initial data

New local smoothing estimates in Besov spaces adapted to the half-wave group are proved via ℓ2-decoupling. We apply these estimates to obtain new well-posedness results for the cubic nonlinear wave equation in two dimensions. The results are compared to new well-posedness results in Lp-based Sobolev spaces.

Degree of convergence of a function of trigonometric series in Besov spaces

In this paper, we study the degree of convergence of the functions of Fourier series and derived Fourier series in Besov spaces using product Hausdorff (HK) means. We also study some applications of our main results.

A PRIORI ESTIMATES FOR THE FIFTH-ORDER MODIFIED KDV EQUATIONS IN BESOV SPACES WITH LOW REGULARITY

We get a priori estimates for the fifth-order modified KdV equations in Besov spaces with low regularity which cover the full subcritical range. These estimates are obtained from the power series expansion of the perturbation determinant associated to the Lax pair. More precisely, we get the global in time bounds of the $B^s_{2,r}$ norm of the solution for $-1/2< s < 1$, $1\leq r\leq \infty$. Then we can obtain the sharp global well-posedness in $H^s$ for $s\geq 3/4$, which is the minimal regularity threshold for which the well-posedness problem can be solved via the contraction principle.

Boundedness from below of composition operators between $L^p_a$ and $L^q_a$, between $L^p_a$ and the Hardy space $H^2$, between $L^p_a$ and Besov space

Boundedness from below of composition operators between $L^p_a$ and $L^q_a$, between $L^p_a$ and the Hardy space $H^2$, between $L^p_a$ and Besov space

Closed range integral operators between L_a^p and L_a^q, between L_a^p and the Hardy space H^2, between L_a^p and Besov space

Closed range integral operators between L_a^p and L_a^q, between L_a^p and the Hardy space H^2, between L_a^p and Besov space