Besov Spaces Research Articles

Extension criterion involving the middle eigenvalue of the strain tensor on local strong solutions to the 3D Navier–Stokes equations

In this article, we prove an extension criterion for a local strong solution to the 3D Navier–Stokes equations that only require control of the positive part of middle eigenvalue of strain tensor in the critical endpoint Besov space, i.e., λ2+∈L2(0,T;Ḃ∞,∞−1). This gives a positive answer to the problem proposed by Miller [1] and improves the results by Wu [2–4]. The proof relies on the identity for enstrophy growth and Lp-norm estimate of the gradient of λ2+.

Navier-Stokes: Singularities and Bifurcations

This article presents substantial advances in the analysis of the Navier-Stokes equations for both compressible and incompressible fluids, focusing on the formation of singularities, hypercomplex bifurcations, and regularity in Sobolev and Besov spaces. Through new theorems, we extend the theory of singularities in fluid dynamics and introduce quaternionic bifurcations, representing an innovative extension of classical bifurcation theory. Moreover, we delve into the investigation of the regularity of compressible fluids, exploring the conditions under which solutions remain smooth or develop singularities. These contributions are fundamental to the understanding of global regularity issues, directly linked to the renowned Millennium Prize problem, which seeks definitive answers on the existence and smoothness of solutions to the Navier-Stokes equations. Additionally, we discuss how these theoretical advancements offer new approaches to unresolved problems related to the formation of singularities in turbulent flows and the multiscale behavior of solutions, which are crucial for a comprehensive understanding of fluid dynamics. This work not only broadens the scope of traditional mathematical analysis of the Navier-Stokes equations but also establishes a robust theoretical framework for the investigation of bifurcations and regularity in advanced functional spaces, fostering a deeper understanding of global regularity phenomena and the complex dynamics governing fluid systems.

The BKM criterion to the 3D double-diffusive magneto convection systems involving planar components

In this paper, we investigate the BKM type blowup criterion applied to 3D double-diffusive magneto convection systems. Specifically, we demonstrate that a unique local strong solution does not experience blow-up at time T, given that ). To prove this, we employ the logarithmic Sobolev inequality in the Besov spaces with negative indices and a well-known commutator estimate established by Kato and Ponce. This result is the further improvement and extension of the previous works by O (2021) and Wu (2023).

Characterization of solutions in Besov spaces for fractional Rayleigh–Stokes equations

This paper considers fractional Rayleigh–Stokes equations with a power-type nonlinearity. The linear equation can be simulated a non-Newtonian fluid for a generalized second grade fluid and display a nonlocal behavior in time. Because the coexistence of fractional and classical derivatives leads to the lack of semigroup structure of the solution operator, we need to develop a suitable tool to establish some Lp−Lq estimates in the framework of Lp spaces and Besov spaces, respectively. Further, global existence of solutions is showed in spaces of Besov type.

A note on the shift theorem for the Laplacian in polygonal domains

We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces.

On the strong summability of the Fourier–Walsh series in the Besov space

UDC 517.5 The Fourier–Walsh series of even continuous functions may be divergent at some points. Moreover, among integrable functions, there are functions such that their Fourier–Walsh series diverge everywhere on [ 0,1 ) . In this connection, it becomes necessary to consider various summation methods that would allow us to restore the function according to its Fourier–Walsh series. We also investigate the Besov space on a dyadic group in terms of strong summability. Finally, we present necessary information about the Fourier–Walsh transform.

Besov Space via Heat Semigroup on Carnot Group and Its Capacity

Besov Space via Heat Semigroup on Carnot Group and Its Capacity

Decomposition and Traces of Weighted Mixed-Norm Besov Spaces

Decomposition and Traces of Weighted Mixed-Norm Besov Spaces

Well-posedness and continuity properties of the Fornberg–Whitham equation in the Besov space $$B^1_{\infty ,1}(\mathbb {R})$$

Well-posedness and continuity properties of the Fornberg–Whitham equation in the Besov space $$B^1_{\infty ,1}(\mathbb {R})$$

Well‐posedness and large‐time behavior for the two‐phase system with magnetic field in critical Besov spaces

AbstractThis paper is dedicated to the study of the two‐phase system with magnetic field in multi‐dimensional spaces . We investigate the local and global well‐posedness of strong solutions to the Cauchy problem in critical Besov spaces. Moreover, a Lyapunov‐type energy argument can be developed, which leads to the time‐decay estimates of solutions without additional smallness assumptions. We improve the previous effects obtained by Wen and Zhu (JDE‐2018), Wang and Cheng (AMS‐2022), where they established the global well‐posedness of strong solutions in the ‐norm Sobolev spaces.

Wavelet and Fourier analytic characterizations of pointwise multipliers of Besov spaces [formula omitted] with 0 < p ≤ 1

For any p∈(0,1] and s∈R, the authors prove two types of characterizations of the pointwise multiplier space M(Bp,ps(Rn)) of the Besov space Bp,ps(Rn). One type is based on wavelet analysis and is an extension of a well-known argument of Yves Meyer. The other type works with Fourier analytic terms. As an application of the above two types of characterizations, the authors further obtain a characterization of bounded functions in the uniform space Bp,p,unifs,τ(Rn) via Haar wavelets in the critical index τ=1p−sn.

Global existence of non-Newtonian incompressible fluid in half space with nonhomogeneous initial-boundary data

In this study, we investigate the global existence of weak solutions of non-Newtonian incompressible fluids governed by (1.1). When u0∈Ḃp,qα−2p(R+n)∩Ḃn+22,n+221−4n+2(R+n)∩Ḃp,11+np(R+n) is given, we will find the weak solutions for the Eq. (1.1) in the function space Cb[0,∞;Ḃp,qα−2p(R+n)∩Cb(0,∞;Ḃn+221−4n+2(R+n))∩L∞(0,∞;Ẇ∞1(R+n)), n+2&amp;lt;p&amp;lt;∞,1≤q≤∞,1+n+2p&amp;lt;α&amp;lt;2. We show the existence of weak solutions in the anisotropic Besov spaces Ḃp,qα,α2(R+n×(0,∞)) (see Theorem 1.2) and we show the embedding Ḃp,qα,α2R+n×(0,∞)⊂Cb[0,∞;Ḃp,qα−2p(R+n) (see Lemma 2.8). For the global existence of solutions, we assume that the extra stress tensor S is represented by S(A)=F(A)A, where F satisfies the assumption (A). Note that S1, S2 and S3 introduced in (1.2) satisfy our assumptions.

On the fractional heat semigroup and product estimates in Besov spaces and applications in theoretical analysis of the fractional Keller–Segel system

This paper is concerned with the fractional Keller–Segel system in temporal and spatial variables. We consider fractional dissipation for the physical variables including a fractional dissipation mechanism for the chemotactic diffusion, as well as a time fractional variation assumed in the Caputo sense. We analyze the fractional heat semigroup obtaining time decay and integral estimates of the Mittag–Leffler operators in critical Besov spaces, and prove a bilinear estimate derived from the nonlinearity of the Keller–Segel system, without using auxiliary norms. We use these results in order to prove the existence of global solutions in critical homogeneous Besov spaces employing only the norm of the natural persistence space, including the existence of self-similar solutions, which constitutes a persistence result in this framework. In addition, we prove a uniqueness result without assuming any smallness condition on the initial data.

On the Ill-posedness for the Navier–Stokes Equations in the Weakest Besov Spaces

On the Ill-posedness for the Navier–Stokes Equations in the Weakest Besov Spaces

Sharp decay characterization for the compressible Navier-Stokes equations

The low-frequency L1 assumption has been extensively applied to the large-time asymptotics of solutions to the compressible Navier-Stokes equations and incompressible Navier-Stokes equations since the classical efforts due to Kawashima, Matsumura, Nishida, Ponce, Schonbek and Wiegner. In this paper, we establish a sharp decay characterization for the compressible Navier-Stokes equations in the critical Lp framework. Precisely, it is proved that the Besov space B˙2,∞σ1-boundedness condition (with d2−2dp≤σ1<d2−1) of the low-frequency part of initial perturbation is not only sufficient, but also necessary to achieve those upper bounds of time-decay estimates. Furthermore, we show that the upper and lower bounds of time-decay estimates hold if and only if the low-frequency part of initial perturbation belongs to a nontrivial subset of B˙2,∞σ1. To the best of our knowledge, our work is the first one addressing the inverse problem for the large-time asymptotics of compressible viscous fluids.

Non-uniform dependence on initial data for the Euler equations in Besov spaces

In this paper, we consider the initial value problem to the higher dimensional Euler equations in the whole space. Based on the local well-posedness result and the lifespan, we prove that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces. Our obtained result improves considerably the previous results given by Himonas-Misiołek (2010) [8] and Pastrana (2021) [21].

Global well-posedness for the three dimensional compressible micropolar equations

In this paper, we study the Cauchy problem of the three-dimensional compressible micropolar equations in the absence of heat-conductivity. By leveraging Fourier theory and employing a refined energy method, we establish the global well-posedness of the equations for small initial data within Besov spaces. As a byproduct, we also derive the optimal time decay of solutions if the low frequency of initial data belonging to Ḃ2,∞−σ1(R3).

Global solutions to the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner system with a class of large initial data

In this paper, we consider the global well-posedness of the three-dimensional inhomogeneous incompressible Phan-Thien–Tanner (PTT) system with general initial data in the critical Besov spaces. The question of whether or not PTT system is globally well posed for large data is still open. When the density ρ is away from zero, we denote by ϱ:=1ρ−1. More precisely, we prove that the PTT system admits a unique global solution, provided that the initial data ϱ 0, the initial horizontal velocity uh0 , the product ωu30 of the coupling parameter ω and the initial vertical velocity u 30, and the initial symmetric tensor of constraints τ 0 are sufficiently small. In particular, this result includes the global well-posedness of the PTT system for small initial data in the case where ω∈[0,1) and for large initial vertical velocity in the case where ω tends to zero. As a by-product, our results can be applied to the so-called Oldroyd-B system.

Norm inequalities with fractional integrals

Fractional spline wavelet systems are considered. Together with the Battle–Lemarié scaling and wavelet functions of natural orders, they are used to find conditions that ensure certain inequalities between the norms of images and pre-images of fractional integration operators in Besov spaces with Muckenhoupt weights on R \mathbb {R} .

Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: Beyond the Lions exponent

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent α can be larger than the Lions exponent 5/4. It is well-known that, due to Lions [1], for any L2 divergence-free initial data, there exist unique smooth Leray-Hopf solutions when α≥5/4. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces LtγWxs,p, in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints (3/p+1−2α,∞,p) and (2α/γ+1−2α,γ,∞). Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff Hη⁎ measure, where η⁎>0 is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.