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This paper aims to understand a blow-up mechanism on a family of shallow-water models with linear dispersion, which are linked with the modified Camassa-Holm equation and the Novikov equation. We first demonstrate the local well-posedness of the model equation in Besov spaces. Our blow-up analysis begins with two cases where the first case is 2 k 1 + 3 k 2 ≠ 0 and then we deduce the results on the curvature blow-up in finite time. To overcome the lack of conservation in the functional due to weak linear dispersion, we can determine a suitable alternative via a slight modification to conserved quantity H 2 [ u ] (see Lemma 4.1). Furthermore, we explore the formation of singularities in another case when nonlocal terms are absent. Lastly, we investigate the Gevrey regularity and analyticity of solutions for Cauchy problem within a specified range of Gevrey-Sobolev spaces by employing the generalized Ovsyannikov theorem and study the continuity of the data-to-solution mapping. [ABSTRACT FROM AUTHOR]

In this paper, we consider the full compressible, viscous, non-resistive MHD system under the assumption that the fluids move on a plane while the magnetic field is oriented vertically. Within the framework of Besov spaces, by introducing several new unknown quantities and exploiting the intrinsic structure of the system, we prove the global well-posedness of strong solutions for initial data close to a constant equilibrium state. Furthermore, under some suitable additional conditions involving only the low-frequency part of the initial perturbation, we develop a Lyapunov-type energy argument, which yields the optimal time-decay rates of the global solution. To the best of our knowledge, our result is the first one on global-in-time solvability to the full compressible, viscous, non-resistive MHD system in multi-dimensional whole space. [ABSTRACT FROM AUTHOR]

In this paper, we prove a global large solution result to the Keller–Segel–Navier–Stokes equations in critical Besov spaces. The result indicates that the global solutions exist without any small conditions imposed on the initial oxygen concentration c 0 . This is different from some known large solutions results, which allow the part large initial data u 0 . [ABSTRACT FROM AUTHOR]

We consider the d-dimensional MagnetoHydroDynamics (MHD) system defined on a sufficiently smooth bounded domain, d = 2 , 3 with homogeneous boundary conditions, and subject to external sources assumed to cause instability. The initial conditions for both fluid and magnetic equations are taken of low regularity. We then seek to uniformly stabilize such MHD system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, static, feedback controls, which are localized on an arbitrarily small interior subdomain. In additional, they will be minimal in number. The resulting space of well-posedness and stabilization is a suitable product space B ~ q , p 2 - 2 / p (Ω) × B ~ q , p 2 - 2 / p (Ω) , 1 < p < 2 q 2 q - 1 , q > d , of tight Besov spaces for the fluid velocity component and the magnetic field component (each "close" to L 3 (Ω) for d = 3 ). Showing maximal L p -regularity up to T = ∞ for the feedback stabilized linear system is critical for the analysis of well-posedness and stabilization of the feedback nonlinear problem. [ABSTRACT FROM AUTHOR]

We construct a new initial data to prove the ill-posedness of both Navier-Stokes and Euler equations in weaker Besov spaces in the sense that the solution maps to these equations starting from u0 are discontinuous at t = 0. [ABSTRACT FROM AUTHOR]

In this paper, we consider the solvability of the two‐dimensional stationary Navier–Stokes equations on the whole plane R2$\mathbb {R}^2$. In Fujii [Ann. PDE, 10 (2024), no. 1. Paper No. 10], it was proved that the stationary Navier–Stokes equations on R2$\mathbb {R}^2$ is ill‐posed for solutions around zero. In contrast, considering solutions around the nonzero constant flow, the perturbed system has a better regularity in the linear part, which enables us to prove the unique existence of solutions in the scaling critical spaces of the Besov type. [ABSTRACT FROM AUTHOR]

In this paper, the estimation of the quantile density function based on i.i.d biased observations is investigated. The bias function is assumed to be positive and bounded. Of the various smoothing methods for selecting the model parameters, hard and block thresholding methods are proposed and two adaptive estimators based on them are constructed. We evaluate these theoretical performances via the minimax approach over Besov balls. We show that these estimators obtain near-optimal and optimal convergence rates under some mild assumptions. Finally, with a simulation study and application on a real set of data, the performance quality of these estimators will be compared to other wavelet methods. [ABSTRACT FROM AUTHOR]

The goal of this work is to investigate finite-time blow-up solutions to a class of time-space fractional diffusion equations with nonlinear exponential source terms. In contrast to the small critical data case, which leads to global solutions, we prove in this study that if the initial Schwartz data is large enough, our solutions will blow up in a finite time. The main idea of the analysis is based on the Fourier analytic approach and embeddings between Triebel-Lizorkin spaces and Besov spaces. [ABSTRACT FROM AUTHOR]

We prove a lower bound estimate for capacities in Hajłasz–Besov, Hajłasz–Triebel–Lizorkin and Hajłasz–Sobolev spaces with generalized smoothness defined on metric spaces in terms of Netrusov–Hausdorff content or Hausdorff content. These results are improvements of the results obtained in [Z. Li, D. Yang and W. Yuan, Lebesgue points of Besov and Triebel–Lizorkin spaces with generalized smoothness, Mathematics 9 2021, 10.3390/math9212724]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial value problems for multi-term time-fractional wave equations in the framework of Besov spaces, which can be described the Couette flow of viscoelastic fluid. Considering the initial data in Besov spaces, we obtain some results about the local well-posedness and the blow-up of mild solutions for the proposed problem. Further, we extend these results to Besov–Morrey spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate singularity formation for the high-order generalized Fokas–Olver–Rosenau–Qiao equation which include the celebrated Fokas–Olver–Rosenau–Qiao equation (or also called the modified Camassa–Holm equation), and we are devoted to understanding how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we show the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, we investigate the formation of singularities and provide sufficient conditions on initial data that lead to the finite time blow-up of the second-order derivative of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D incompressible magnetohydrodynamic equations and establish several logarithmically improved regularity criteria in the homogeneous Besov space. These criteria are based on certain assumptions made in the velocity field and pressure, respectively. The first result solely based on the velocity field indicates that the velocity field plays a dominant role in determining the regularity of weak solutions, when compared with the magnetic field. [ABSTRACT FROM AUTHOR]

We investigate the nonparametric additive regression estimation in random design and long-memory errors and construct adaptive thresholding estimators based on wavelet series. The proposed approach achieves asymptotically near-optimal convergence rates when the unknown function and its univariate additive components belong to Besov space. We consider the problem under two noise structures; (1) homoskedastic Gaussian long memory errors and (2) heteroskedastic Gaussian long memory errors. In the homoskedastic long-memory error case, the estimator is completely adaptive with respect to the long-memory parameter. In the heteroskedastic long-memory case, the estimator may not be adaptive with respect to the long-memory parameter unless the heteroskedasticity is of polynomial form. In either case, the convergence rates depend on the long-memory parameter only when long-memory is strong enough, otherwise, the rates are identical to those under i.i.d. errors. In addition, convergence rates are free from the curse of dimensionality. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a class of Besov spaces via the heat semigroup on Carnot groups, and provide some basic properties about these spaces. Especially, we reveal the relationship between the Besov space and the known Sobolev space on Carnot groups. Then some Sobolev type inequalities for Besov spaces are established. Finally, we introduce the Besov capacity associated with Besov spaces and investigate some measure theories and geometric properties for this capacity. Furthermore, several applications of the Besov capacity are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the local well-posedness for the Fornberg–Whitham-type equation in the Besov spaces B p , r s (R) with 1 ≤ p , r ≤ ∞ and s > m a x { 1 + 1 p , 3 2 } , which improve the previous work in Sobolev spaces H s (R) = B 2 , 2 s (R) with s > 3 2 (Lai and Luo in J Differ Equ 344:509–521, 2023). Furthermore, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces B p , r s (R) with 1 ≤ p ≤ ∞ , 1 ≤ r < ∞ and s > m a x { 1 + 1 p , 3 2 } . [ABSTRACT FROM AUTHOR]

Very recently, Boutros and Titi considered Onsager's conjecture of α$\alpha$‐type Euler models in Besov spaces B3,∞β$B^{\beta }_{3,\infty }$ and B3,∞γ$B^{\gamma }_{3,\infty }$ with β>0,γ>1$\beta >0, \gamma >1$. In this paper, we study the energy conservation of weak solutions to these systems in Lebesgue space L3(T3)$L^{3}(\mathbb {T}^{3})$ and homogenous Sobolev space Ẇ1,3(T3)$\dot{W}^{1,3}(\mathbb {T}^{3})$. Moreover, the corresponding results for the energy and the cross‐helicity of the inviscid α$\alpha$‐magnetohydrodynamic equations are also obtained. [ABSTRACT FROM AUTHOR]

Abstract.Deconvolution model is investigated when the design points ti are random and follow a bounded density h. We propose a wavelet hard-thresholding estimator, and show that it achieves asymptotically near-optimal convergence rates for the L2-risk when the response function f belongs to Besov space. We consider the problem under two noise structure; (1) i.i.d. Gaussian errors and (2) long-memory Gaussian errors. At a first stage, we assume h is known. For the i.i.d. Gaussian errors case, our convergence rates are similar to those found in the literature under regular design. For the long memory case, our convergence rates are new. However, at each elbow, the convergence rates will depend on the long-memory parameter only if it is small enough, otherwise, the rates will be identical to those we obtained under i.i.d. errors. At a second stage, we assume h is unknown, construct the usual hard-thresholding estimator, and derive the upper bounds for its L2-risk under the new settings. The rates seem to depend on the smoothness parameter, γ, of h when γ is relatively small. [ABSTRACT FROM AUTHOR]

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, Ω−⊂ℝ3$$ {\Omega}^{-}\subset {\mathrm{\mathbb{R}}}^3 $$, has a compact boundary surface S=∂Ω−$$ S=\partial {\Omega}^{-} $$, which is divided into two disjoint parts, the Dirichlet part SD$$ {S}_D $$ and the Neumann part SN$$ {S}_N $$, where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate L2$$ {L}_2 $$‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space [W2,loc1(Ω−)]3$$ {\left[{W}_{2, loc}^1\left({\Omega}^{-}\right)\right]}^3 $$ and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the Lp$$ {L}_p $$‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses Cα$$ {C}^{\alpha } $$‐Hölder continuity property in the closed domain Ω−‾$$ \overline{\Omega^{-}} $$ with α=12−ε$$ \alpha =\frac{1}{2}-\varepsilon $$, where ε>0$$ \varepsilon >0 $$ is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

A new wavelet-based estimator of the conditional density is investigated. The estimator is constructed by combining a special ratio technique and applying a non negative estimator to the density function in the denominator. We used a wavelet shrinkage technique to find an adaptive estimator for this problem. In particular, a block thresholding estimator is proposed, and we prove that it enjoys powerful mean integrated squared error properties over Besov balls. Moreover, it is shown that convergence rates for the mean integrated squared error (MISE) of the adaptive estimator are optimal under some mild assumptions. Finally, a numerical example has been considered to illustrate the performance of the estimator. [ABSTRACT FROM AUTHOR]

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]. [ABSTRACT FROM AUTHOR]

Under the singular initial conditions, we consider the parabolic Anderson model driven by the time-independent generalized Gaussian fields, which include some frequently-used non-stationary Gaussian fields. When the initial value u0 belongs to the weighted Besov space with negative regularity B q , ∞ − κ , w λ , it is proved that there exists a unique pathwise solution for the model in the Young sense. Moreover, if u0 also satisfies the measure-valued initial condition, by the Feynman-Kac formula based on Brownian bridge, we find that the solution owns a spatially Hölder continuous modification. [ABSTRACT FROM AUTHOR]

We study a nonlocal abstract Ginzburg–Landau type equation. The equation includes variable coefficients with convolution terms and an abstract linear operator function in a Fourier-type Banach space . For sufficiently smooth initial data, assuming growth conditions for the operator and the coefficient , the existence and uniqueness of the solution and the -regularity properties are established. We obtain the existence and uniqueness of the solution, and the regularity of different classes of nonlocal Ginzburg–Landau-type equations by choosing the space and operator that occur in a wide variety of physical systems. [ABSTRACT FROM AUTHOR]

In this paper, we first give characterizations of weighted Besov spaces with variable exponents via Peetre's maximal functions. Then we obtain decomposition characterizations of these spaces by atom, molecule and wavelet. As an application, we obtain the boundedness of the pseudo-differential operators on these spaces. [ABSTRACT FROM AUTHOR]

In this paper we derive caloric smoothing estimates in Morrey smoothness spaces using decomposition techniques by means of wavelets and molecules. Our new estimate extends results for Gauss-Weierstrass, Cauchy-Poisson and fractional Gauss-Weierstrass semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we mainly focus on studying the multi-dimensional compressible Oldroyd-B model, which is derived by Barrett, Lu, and Süli (Comm. Math. Sci. 2017) through the micro-macro analysis of the compressible Navier-Stokes-Fokker-Planck system in the case of Hookean bead-spring chains. We provide a unified method to study the system with the background polymer number density $ \eta_{\infty}\ge 0 $, including the vanishing case and the nonvanishing case, and establish the global-in-time existence of the strong solution for the associated Cauchy problem when the initial data are small in the critical Besov spaces. Comparing Zhai-Li's result (J. Differential Equations 2021), we consider the system with the multi-dimensional case $ n\geq2 $ and the background polymer number density $ \eta_{\infty}\ge 0 $, not only $ n = 2,3 $ and $ \eta_{\infty} = 0 $. And we also extend the initial condition for the global existence of strong solutions by Zhai and Li (J. Differential Equations 2021) to a large class of the initial data, specifically, $ (\eta_0-\eta_{\infty})^l\in \dot{B}^{\frac{n}{2}-1}_{2,1}(\mathbb{R}^n) $ and $ (\mathbb{T}_0-\mathbb{T}_{\infty})^h\in\dot{B}^{\frac{n}{p}-1+s}_{p,1}(\mathbb{R}^n) $ with $ -1\le s\le 0 $ and $ s>1-\frac{2n}{p} $. Additionally, we also obtain the decay rates of the solution in Besov spaces. [ABSTRACT FROM AUTHOR]

The aim of this study was to establish a new regularity criterion for local-in-time smooth solutions for the 3D nematic liquid crystal flows in terms of the horizontal gradient of two horizontal velocity components in the framework of the homogeneous Besov Ḃ ∞ , ∞ − 1 space and one directional derivative of molecular orientations in the framework of the homogeneous Morrey–Campanato Ṁ 2 , 3 r space. More precisely, we show that the smooth solution (u , d) can be extended beyond T , provided that ∫ 0 T (∥ ∇ h u ̃ (⋅ , t) ∥ B ̇ ∞ , ∞ − 1 2 + ∥ ∂ 3 d (⋅ , t) ∥ Ṁ 2 , 3 r 2) d t < ∞. [ABSTRACT FROM AUTHOR]

In this paper, we establish a blow-up criterion for the local smooth solutions to the three-dimensional incompressible Boussinesq equations in the framework of Besov spaces $ \dot{B}_{\infty ,\infty }^{0} $ via partial horizontal derivatives of two velocity components $ \partial _{1}u_{1} $ and $ \partial _{2}u_{2} $. Our result can be regarded as the borderline case of the work by F. Wu (Bull. Braz. Math. Soc., New Series (52) 2021, 267-279). [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

This work is concerned with the global existence of large solutions to the three‐dimensional dissipative fluid‐dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the incompressible Navier–Stokes–Poisson–Nernst–Planck equations. Making full use of the algebraic structure of the system, we obtain the global existence of solutions without smallness assumptions imposed on the third component of the initial velocity field and the summation of initial densities of charged species. More precisely, we prove that there exist two positive constants c0,C0$$ {c}_0,{C}_0 $$ such that if the initial data satisfies u0hB˙p,1−1+3p+‖N0−P0‖B˙q,1−2+3qexpC0u03B˙p,1−1+3p2+‖N0+P0‖B˙r,1−2+3r+1expC0u03B˙p,1−1+3p+1≤c0,$$ {\displaystyle \begin{array}{ll}\left({\left\Vert {u}_0^h\right\Vert}_{{\dot{B}}_{p,1}^{-1+\frac{3}{p}}}+{\left\Vert {N}_0-{P}_0\right\Vert}_{{\dot{B}}_{q,1}^{-2+\frac{3}{q}}}\right)& \exp \left\{{C}_0\left({\left\Vert {u}_0^3\right\Vert}_{{\dot{B}}_{p,1}^{-1+\frac{3}{p}}}^2+\left({\left\Vert {N}_0+{P}_0\right\Vert}_{{\dot{B}}_{r,1}^{-2+\frac{3}{r}}}+1\right)\right.\right.\\ {}& \left.\left.\exp \left\{{C}_0{\left\Vert {u}_0^3\right\Vert}_{{\dot{B}}_{p,1}^{-1+\frac{3}{p}}}\right\}+1\right)\right\}\le {c}_0,\end{array}} $$then the incompressible Navier–Stokes–Poisson–Nernst–Planck equations admits a unique global solution. [ABSTRACT FROM AUTHOR]

The local existence of solutions to nonhomogeneous Navier–Stokes equations in cylindrical domains with arbitrary large flux is demonstrated. The existence is proved by the method of successive approximations. To show the existence with the lowest possible regularity the special Besov spaces called the Sobolev–Slobodetskii spaces are used. The inflow and outflow are prescribed on the parts of the boundary which are perpendicular to the x3$$ {x}_3 $$‐axis. Since the inflow and outflow are positive the crucial point of this paper is to verify that x3$$ {x}_3 $$‐coordinate of velocity is also positive. Finally, we conclude the local existence such that the velocity belongs to Wσ2+s,1+s/2(Ωt)$$ {W}_{\sigma}^{2+s,1+s/2}\left({\Omega}^t\right) $$, the gradient of pressure to Wσs,s/2(Ωt)$$ {W}_{\sigma}^{s,s/2}\left({\Omega}^t\right) $$ and the density to Wr,∞1,1(Ωt)$$ {W}_{r,\infty}^{1,1}\left({\Omega}^t\right) $$, where s∈(0,1),σ>3/s,r>5/s,r>σ$$ s\in \left(0,1\right),\sigma >3/s,r>5/s,r>\sigma $$. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove that the sequence spaces ℓ p ⊕ ℓ q and the spaces of infinite matrices ℓ p (ℓ q) , ℓ q (ℓ p) and (⨁ n = 1 ∞ ℓ p n) ℓ q , which are isomorphic to certain Besov spaces, have an almost greedy basis whenever 0 < p < 1 < q < ∞ . More precisely, we custom-build almost greedy bases in such a way that the Lebesgue parameters grow in a prescribed manner. Our arguments critically depend on the extension of the Dilworth–Kalton–Kutzarova method from Dilworth et al. (Stud Math 159(1):67–101, 2003), which was originally designed for constructing almost greedy bases in Banach spaces, to make it valid for direct sums of mixed-normed spaces with nonlocally convex components. Additionally, we prove that the fundamental functions of all almost greedy bases of these spaces grow as (m 1 / q) m = 1 ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we focus on the well‐posedness, blow‐up phenomena, and continuity of the data‐to‐solution map of the Cauchy problem for a two‐component higher order Camassa–Holm (CH) system. The local well‐posedness is established in Besov spaces Bp,11p×Bp,12+1p$B_{p,1}^{\frac{1}{p}} \times B_{p,1}^{2+\frac{1}{p}}$ with 1≤p<∞$1 \le p < \infty$, which improves the local well‐posedness result proved before in Tang and Liu [Z. Angew. Math. Phys. 66 (2015), 1559–1580], Ye and Yin [arXiv preprint arXiv:2109.00948 (2021)], Zhang and Li [Nonlinear Anal. Real World Appl. 35 (2017), 414–440], and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. Next, we consider the continuity of the solution‐to‐data map, that is, the ill‐posedness is derived in Besov space Bp,∞s−2×Bp,∞s$B_{p,\infty }^{s - 2} \times B_{p,\infty }^s$ with 1≤p≤∞$1 \le p \le \infty$ and s>max{2+1p,52}$s>\max \lbrace 2+\frac{1}{p},\frac{5}{2}\rbrace$. Then, the nonuniform continuous and Hölder continuous dependence on initial data for this system are also presented in Besov spaces Bp,rs−2×Bp,rs$B_{p,r}^{s - 2} \times B_{p,r}^s$ with 1≤p,r<∞$1 \le p,r < \infty$ and s>max{2+1p,52}$s > \max \lbrace {2+\frac{1}{p},\frac{5}{2}}\rbrace$. Finally, the precise blow‐up criteria for the strong solutions of the two‐component higher order CH system is determined in the lowest Sobolev space Hs−2×Hs$H^{s-2}\times H^s$ with s>52$s>\frac{5}{2}$, which improves the blow‐up criteria result established before in He and Yin [Discrete Contin. Dyn. Syst. 37 (2016), no. 3, 1509–1537] and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. [ABSTRACT FROM AUTHOR]

In this paper, we mainly concern with the system of the coupled nonlinear Schrödinger equation with initial data in critical spaces in dimension 3. We first obtain the global well-posedness of the solution to the coupled nonlinear Schrödinger equation with initial data in a critical space $ W^{\frac {11}{7}, \frac 76}({\mathbb {R}}^3) $ W 11 7 , 7 6 ( R 3). The key is to derive a uniform bound of a modified energy $ \mathcal {E}(t) $ E (t) based on a decomposition for the solution as in Dodson [Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space. Rev Mat Iberoam. 2022;38:1087–1100]. In addition, inspired by Dodson [Scattering for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space. International mathematics research notices; 2023], we make a new decomposition for the solution and show the scattering to the coupled nonlinear Schrödinger equation with radially symmetric initial data in a critical Besov space $ B_{1,1}^2({\mathbb {R}}^3) $ B 1 , 1 2 ( R 3). [ABSTRACT FROM AUTHOR]

It was proved in Iwabuchi and Ogawa (J Elliptic Parabol Equ 7(2):571–587, 2021) that the Cauchy problem for the full compressible Navier–Stokes equations of the ideal gas is ill-posed in B ˙ p , q 2 / p (R 2) × B ˙ p , q 2 / p - 1 (R 2) × B ˙ p , q 2 / p - 2 (R 2) with 1 ≤ p ≤ ∞ and 1 ≤ q < ∞ . In this paper, we aim to solve the end-point case left in [17] and prove that the Cauchy problem is ill-posed in B ˙ p , ∞ d / p (R d) × B ˙ p , ∞ d / p - 1 (R d) × B ˙ p , ∞ d / p - 2 (R d) with 1 ≤ p ≤ ∞ by constructing a sequence of initial data for showing discontinuity of the solution map at zero. As a by-product, we demonstrate that the Cauchy problem for the incompressible Navier–Stokes equations is also ill-posed in B ˙ p , ∞ d / p - 1 (R d) , which is an interesting open problem in itself. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the study of the two‐phase system with magnetic field in multi‐dimensional spaces d≥2$d\ge 2$. 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 H2(R3)$ H^{2}(\mathbb {R}^{3})$‐norm Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to the study of the compressible viscous Baer-Nunziato (BN) system in multi-dimensional spaces with $ d\geq2 $. Compared to previous findings, the (BN) system for compressible two-phase flows is led in two pressure state laws that vary with different phases. This property adds a practical physics background to this model. We will investigate the global existence of strong solutions to the Cauchy problem within the $ L^p $ critical regularity framework. Furthermore, we develop a Lyapunov-type energy argument that provides time-decay estimates of solutions without requiring additional smallness assumptions. [ABSTRACT FROM AUTHOR]

In the paper, we consider the regularity of the weak solutions for the incompressible 3D Navier–Stokes (N–S) equations. Our main result is the double‐logarithmic regularity criterion of pressure for the 3D Navier–Stokes equations in the Besov space. ∫0TπB˙∞,∞−3/qp/e+lne+πB˙∞,∞−3/qlne+lne+πB˙∞,∞−3/qdt<∞, 2323/q+/q=,≤q≤∞. Our results extend and generalize previous works. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with global well-posedness and optimal decay rate for strong solutions of the compressible finite extensible nonlinear elastic (FENE) dumbbell model. For d ≥ 2 , we firstly prove that the compressible FENE dumbbell model admits the unique global strong solutions provided initial data are close to equilibrium state. Then, by the Littlewood-Paley decomposition theory and the Fourier splitting method, we show optimal L 2 decay rate of global strong solutions for d ≥ 3. Finally, we mainly study optimal decay rate to the 2-D FENE dumbbell model. The improved Fourier splitting method yields that the L 2 decay rate is ln − l ⁡ (e + t) for any l ∈ N. By virtue of the time weighted energy estimate, we can improve the decay rate to (1 + t) − 1 4 . Under the low-frequency condition and by the Littlewood-Paley theory, we show that the solutions belong to some Besov spaces with negative index and obtain optimal L 2 decay rate without the smallness restriction of low frequencies. [ABSTRACT FROM AUTHOR]

We consider the global well-posedness and decay rates for solutions of 3D incompressible micropolar equation in the critical Besov space. Spectrum analysis allows us to find not only parabolic behaviors of solutions, but also damping effect of angular velocity in the low frequencies. Based on this observation, we establish the global well-posedness with more general regularity on the initial data. The approach concerns large time behaviors is so-called the Gevrey method which bases on the parabolic mechanics of the micropolar system. This method enables us even to derive decay of arbitrary higher order derivatives by transforming it into the control of the radius of analyticity under a particular regularity. To bound the growth of the radius of analyticity in general L p Besov spaces, we shall develop some new techniques concern Gevrey estimates, especially an extended Coifman-Meyer theory, to cover those endpoint Lebesgue framework. [ABSTRACT FROM AUTHOR]

This article considers wavelet estimations for density functions in Besov spaces based on negatively dependent biased random samples. More precisely, wavelet estimators are defined and their L p (1 ≤ p < ∞) convergence rates are provided when the parameter μ is known and unknown, respectively. [ABSTRACT FROM AUTHOR]

In this study, we investigate the global existence of weak solutions of non-Newtonian incompressible fluids governed by (1.1). When u 0 ∈ B ̇ p , q α − 2 p ( R + n ) ∩ B ̇ n + 2 2 , n + 2 2 1 − 4 n + 2 ( R + n ) ∩ B ̇ p , 1 1 + n p ( R + n ) is given, we will find the weak solutions for the Eq. (1.1) in the function space C b [ 0 , ∞ ; B ̇ p , q α − 2 p ( R + n ) ∩ C b (0 , ∞ ; B ̇ n + 2 2 1 − 4 n + 2 ( R + n )) ∩ L ∞ (0 , ∞ ; W ̇ ∞ 1 ( R + n )) , n + 2 < p < ∞ , 1 ≤ q ≤ ∞ , 1 + n + 2 p < α < 2. We show the existence of weak solutions in the anisotropic Besov spaces B ̇ p , q α , α 2 ( R + n × (0 , ∞)) (see Theorem 1.2) and we show the embedding B ̇ p , q α , α 2 R + n × (0 , ∞) ⊂ C b [ 0 , ∞ ; B ̇ p , q α − 2 p ( 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. [ABSTRACT FROM AUTHOR]

This paper investigates the bipolar non-isentropic compressible Euler-Maxwell system in $ \mathbb {R}^{3} $ R 3 and $ \mathbb {T}^{3} $ T 3 . For both problems, we establish the global existence of smooth solutions in the general Besov spaces, which covers the usual Sobolev spaces with higher regularity and the critical Besov space, when the initial perturbations around the constant states are small enough. As a byproduct, we obtain the large-time asymptotic behavior of the global solutions near the equilibrium state in the general Besov spaces with relatively lower regularity. The proof is based on the technical Fourier frequency-localization method developed through the Littlewood-Paley theory, but some new development and technique are proposed for treating the strong coupling and nonlinearity for the bipolar non-isentropic case. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem of one-dimensional compressible Euler–Fourier–Korteweg system. The global unique strong solutions are established in the critical Besov spaces with small initial data close to a constant equilibrium state. This extends the recent work of Kawashima et al. (Commun Partial Differ Equ 47:378–400, 2022) on the dissipative structure of linear Euler–Fourier–Korteweg system to the non-linear system in critical space. [ABSTRACT FROM AUTHOR]

We consider stochastic differential equations (SDEs) with (distributional) drift in negative Besov spaces and random initial condition and investigate them from two different viewpoints. In the first part we set up a martingale problem and show its well-posedness. We then prove further properties of the martingale problem, such as continuity with respect to the drift and the link with the Fokker–Planck equation. We also show that the solutions are weak Dirichlet processes for which we evaluate the quadratic variation of the martingale component. In the second part we identify the dynamics of the solution of the martingale problem by describing the proper associated SDE. Under suitable assumptions we show equivalence with the solution to the martingale problem. [ABSTRACT FROM AUTHOR]

In this paper, our primary focus is to study the boundedness and compactness of Volterra type operators and multiplication operators on minimal α -Möbius invariant function spaces. Additionally, we also present a characterization of the boundedness and compactness of Volterra type and multiplication operators from minimal α -Möbius invariant function spaces to Besov spaces. [ABSTRACT FROM AUTHOR]

This paper establishes non-uniform continuity of the data-to-solution map in the periodic case for the two-component Fornberg-Whitham system in Besov spaces Bs p,r (T) x B s−1 p,r (T) for s > max{2 + 1 p, 5 2 }. In particular, when p = 2 and r = 2, this proves the non-uniform dependence on initial data for the system in Sobolev spaces Hs (T) × H s−1 (T) for s > 5/2 . [ABSTRACT FROM AUTHOR]

We study a class of almost diagonal matrices compatible with the mixed-norm α-modulation spaces Mp,qs,α(ℝn), α ∈ [0, 1], introduced recently by Cleanthous and Georgiadis. The class of almost diagonal matrices is shown to be closed under matrix multiplication and we connect the theory to the continuous case by identifying a suitable notion of molecules for the mixed-norm α-modulation spaces. It is shown that the “change of frame” matrix for a pair of time-frequency frames for the mixed-norm α-modulation consisting of suitable molecules is almost diagonal. As examples of applications, we use the almost diagonal matrices to construct compactly supported frames for the mixed-norm α-modulation spaces, and to obtain a straightforward boundedness result for Fourier multipliers on the mixed-norm α-modulation spaces. [ABSTRACT FROM AUTHOR]

The paper is concerned with the three-dimensional Boussinesq-Coriolis equations with Caputo time-fractional derivatives. Specifically, by striking new balances between the dispersion effects of the Coriolis force and the smoothing effects of the Laplacian dissipation involving with a time-fractional evolution mechanism, we obtain the global existence of mild solutions to Cauchy problem of three-dimensional time-fractional Boussinesq-Coriolis equations in Besov spaces. [ABSTRACT FROM AUTHOR]