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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, 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]

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]

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]

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]

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]

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 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]

We study the regularity properties of Hölder continuous minimizers to non‐autonomous functionals satisfying (p,q)$(p,q)$‐growth conditions, under Besov assumptions on the coefficients. In particular, we are able to prove higher integrability and higher differentiability results for solutions to our minimum problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the parabolic–elliptic Keller–Segel system, which is coupled to the incompressible Navier–Stokes equations through transportation and friction. It is shown that when the system is diffused by Lévy motion, the well‐posedness of the mild solution to the corresponding Cauchy problem in homogeneous Besov spaces is established by means of the Banach fixed point theorem. Furthermore, we prove the Lorentz regularity in time direction and the maximal regularity of solutions. In addition, we obtain the additional regularity and explore the time decay property of global mild solutions. [ABSTRACT FROM AUTHOR]

We study stochastic reaction–diffusion equation ∂tut(x)=12∂xx2ut(x)+b(ut(x))+Ẇt(x),t>0,x∈D$$\begin{equation*} \partial _tu_t(x)=\frac{1}{2} \partial ^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in {D} \end{equation*}$$where b$b$ is a generalized function in the Besov space Bq,∞β(R)$\mathcal {B}^\beta _{q,\infty }(\mathbb {R})$, D⊂R${D}\subset \mathbb {R}$ and Ẇ$\dot{W}$ is a space‐time white noise on R+×D$\mathbb {R}_+\times {D}$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever β−1/q⩾−1$\beta -1/q\geqslant -1$, β>−1$\beta >-1$ and q∈[1,∞]$q\in [1,\infty ]$. This class includes equations with b$b$ being measures, in particular, b=δ0$b=\delta _0$ which corresponds to the skewed stochastic heat equation. For β−1/q>−3/2$\beta -1/q > -3/2$, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalize the results of Gyöngy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê (2020). [ABSTRACT FROM AUTHOR]

We study the well‐posedness of the degenerate fractional differential equations with finite delay: Dα(Mu)(t)+cDβ(Mu)(t)$D^\alpha (Mu)(t) + cD^\beta (Mu)(t)$=Au(t)+Fut+f(t),(0≤t≤2π)$= Au(t) + Fu_t + f(t),(0\le t\le 2\pi)$ on Lebesgue–Bochner spaces Lp(T;X)$L^p(\mathbb {T}; X)$ and periodic Besov spaces Bp,qs(T;X)$B_{p,q}^s(\mathbb {T}; X)$, where A$A$ and M$M$ are closed linear operators in a complex Banach space X$X$ satisfying D(A)⊂D(M)$D(A)\subset D(M)$, c∈C$c\in \mathbb {C}$ and 0<β<α$0 < \beta < \alpha$ are fixed, ut(s)=u(t+s)$u_t(s) = u(t+s)$ when 0≤t≤2π,−2π≤s≤0$0\le t\le 2\pi , -2\pi \le s\le 0$, and the delay operator F$F$ is a bounded linear operator from Lp([−2π,0];X)$L^p([-2\pi ,0]; X)$ (resp. Bp,qs([−2π,0];X)$B_{p,q}^s([-2\pi ,0]; X)$) into X$X$. Using known operator‐valued Fourier multiplier theorems on Lp(T;X)$L^p(\mathbb {T}; X)$ and Bp,qs(T;X)$B_{p,q}^s(\mathbb {T}; X)$, we completely characterize the Lp$L^p$‐well‐posedness and the Bp,qs$B_{p,q}^s$‐well‐posedness of above equations. We also give concrete examples that our abstract results may be applied. [ABSTRACT FROM AUTHOR]

Given a bounded Lipschitz domain Ω⊂Rn$\Omega \subset \mathbb {R}^n$, Rychkov showed that there is a linear extension operator E$\mathcal {E}$ for Ω$\Omega$, which is bounded in Besov and Triebel‐Lizorkin spaces. In this paper, we introduce some new estimates for the extension operator E$\mathcal {E}$ and give some applications. We prove the equivalent norms ∥f∥Apqs(Ω)≈∑|α|≤m∥∂αf∥Apqs−m(Ω)$\Vert f\Vert _{\mathcal A_{pq}^s(\Omega)}\approx \sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathcal A_{pq}^{s-m}(\Omega)}$ for general Besov and Triebel‐Lizorkin spaces. We also derive some quantitative smoothing estimates of the extended function and all its derivatives on Ω¯c$\overline{\Omega }^c$ up to the boundary. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the continuity of the solution map for the Cauchy problem of the Whitham equation. First, the continuity dependence of solution is established in B2,rs$\mathrm{B}^{s}_{2,r}$ in the sense of Hadamard. Next, by constructing approximate solutions, we show that the data‐to‐solution map is not uniformly continuous in Besov spaces B2,rs$\mathrm{B}^{s}_{2,r}$ (s>32,1≤r≤∞$s>\frac{3}{2}, 1\le r\le \infty$) on the periodic case and on the line. The crucial technical tool used to prove this result is Lemma 5.1, which generalizes the result of Lemma 3 in [23]. [ABSTRACT FROM AUTHOR]

In this paper, the local well‐posedness (which means that the initial‐to‐solution map φ↦u$$ \varphi \mapsto u $$ is existence, uniqueness and continuous) for the Cauchy problem of the Keller‐Segel model with nonlinear chemotactic sensitivity and signal secretion in Besov spaces Bp,rs(ℝd)$$ {B}_{p,r}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$ with 1≤p≤+∞,1≤r<+∞$$ 1\le p\le &#x0002B;\infty, 1\le r&lt;&#x0002B;\infty $$ and s>1+dp$$ s&gt;1&#x0002B;\frac{d}{p} $$ was established, and the solution of PEKS (Keller‐Segel system with β>0$$ \beta &gt;0 $$) converges to the solution of HEKS (Keller‐Segel system with β=0$$ \beta &#x0003D;0 $$) as the diffusion coefficient β→0$$ \beta \to 0 $$ in these Besov spaces was proved. In addition, we show that the solution of this parabolic‐elliptic chemotaxis‐growth system is local well‐posedness in the critical spaces Bp,11+dp(ℝd)$$ {B}_{p,1}&#x0005E;{1&#x0002B;\frac{d}{p}}\left({\mathbb{R}}&#x0005E;d\right) $$ but ill‐posed (not continuous) in Besov spaces B2,∞s(ℝd)$$ {B}_{2,\infty}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$. Moreover, we show a further continuity result that the initial‐to‐solution map φ↦u$$ \varphi \mapsto u $$ is Hölder continuous in Besov spaces Bp,rs(ℝd)$$ {B}_{p,r}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$ equipped with weaker topology. Finally, a blow‐up criteria for the solutions of this system in Besov spaces Bp,rs(ℝd)$$ {B}_{p,r}&#x0005E;s\left({\mathbb{R}}&#x0005E;d\right) $$ was also obtained. [ABSTRACT FROM AUTHOR]

We consider the composition operators Tf:g↦f∘g$T_f: g\mapsto f\circ g$ acting on the real‐valued homogeneous Besov or Triebel–Lizorkin spaces, realized as dilation invariant subspaces of S′(Rn)$\mathcal {S}^{\prime }({\mathbb {R}}^n)$, denoted as Ap,qs(Rn)$\mathfrak {A}^s_{p,q}({\mathbb {R}}^n)$. If s>1+(1/p)$s>1+ (1/p)$ and s≠n/p$s\not= n/p$, then any function f:R→R$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ acting by composition on Ap,qs(Rn)$\mathfrak {A}^s_{p,q}({\mathbb {R}}^n)$ is necessarily linear. The above conditions are optimal: (i) in case s=n/p$s=n/p$, 0

Modified from the standard half‐space extension via the reflection principle, we construct a linear extension operator for the upper half space R+n$\mathbb {R}^n_+$ that has the form Ef(x)=∑j=−∞∞ajf(x′,−bjxn)$Ef(x)=\sum _{j=-\infty }^\infty a_jf(x^{\prime },-b_jx_n)$ for xn<0$x_n<0$. We prove that E$E$ is bounded in all Ck$C^k$‐spaces, Sobolev and Hölder spaces, Besov and Triebel–Lizorkin spaces, along with their Morrey generalizations. We also give an analogous construction on bounded smooth domains. [ABSTRACT FROM AUTHOR]

This paper focuses on Chemin's space Sh′$\mathcal {S}^{\prime }_h$ of homogeneous distributions, which was introduced to serve as a basis for the realizations of subcritical homogeneous Besov spaces. We will discuss how this construction fails in multiple ways for supercritical spaces. In particular, we study its intersection Xh:=Sh′∩X$X_h := \mathcal {S}^{\prime }_h \cap X$ with various Banach spaces X$X$, namely supercritical homogeneous Besov spaces and the Lebesgue space L∞$L^\infty$. For each X$X$, we investigate whether the intersection Xh$X_h$ is dense in X$X$. If it is not, then we study its closure C=clos(Xh)$C = {\rm clos}(X_h)$ and prove that the quotient X/C$X/C$ is not separable and that C$C$ is not complemented in X$X$. [ABSTRACT FROM AUTHOR]

We study the problem where a one‐dimensional elastic string is immersed in a two‐dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non‐linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time wellposedness for arbitrary initial data in the scaling critical Besov space Ḃ2,13/2(T;R2)$\dot{B}^{3/2}_{2,1}(\mathbb {T}; \mathbb {R}^2)$. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancelation structure. [ABSTRACT FROM AUTHOR]

In this paper, we focus on Cauchy problem of the non‐isentropic compressible Navier–Stokes–Poisson system with the initial perturbation (ρ0−ρ¯,m0,θ0−θ¯)$$ \left({\rho}&#x0005E;0-\overline{\rho},{\mathbf{m}}&#x0005E;0,{\theta}&#x0005E;0-\overline{\theta}\right) $$ belonging to the space Hl(ℝ3)∩B˙1,∞−s(ℝ3)$$ {H}&#x0005E;l\left({\mathrm{\mathbb{R}}}&#x0005E;3\right)\cap {\dot{B}}_{1,\infty}&#x0005E;{-s}\left({\mathrm{\mathbb{R}}}&#x0005E;3\right) $$, where l⩾4$$ l\geqslant 4 $$ and s∈[0,1]$$ s\in \left[0,1\right] $$. More importantly, we establish the optimal temporal decay rate of the global strong solution, which can be considered as further work. [ABSTRACT FROM AUTHOR]

We consider K‐interpolation spaces involving slowly varying functions, and derive necessary and sufficient conditions for a Holmstedt‐type formula to be held in the limiting case θ0=θ1∈{0,1}$\theta _0=\theta _1\in \lbrace 0,1\rbrace$. We also study the case θ0=θ1∈(0,1)$\theta _0=\theta _1\in (0,1)$. Applications are given to Lorentz–Karamata spaces, generalized gamma spaces, and Besov spaces. [ABSTRACT FROM AUTHOR]

This work deals with well‐posedness and blow‐up in the setting of Lebesgue and Besov spaces to the time‐fractional Keller–Segel model for chemotaxis under homogeneous Neumann boundary conditions in a smooth domain of RN$\mathbb {R}^N$. The KS model consists in a coupled system of partial differential equations. In particular, we also treat the unique continuation of the solution and the persistence of continuous dependence on the initial data for the continued solution. [ABSTRACT FROM AUTHOR]

Considered herein is the Cauchy problem for a four‐component Novikov system with peaked solutions. We first investigate the local Gevrey regularity and analyticity of the solutions by a generalized Ovsyannikov theorem. Then, based on the local well‐posedness of this problem, the results with respect to the nonuniformly continuous dependence on initial data of the solutions in Besov spaces B2,15/2(핋)2×B2,13/2(핋)2 and Bp,rs(ℝ)2×Bp,rs−1(ℝ)2(s>max{5/2,2+1/p},1≤p,r≤∞)$$ {\left({B}_{p,r}&amp;#x0005E;s\left(\mathrm{\mathbb{R}}\right)\right)}&amp;#x0005E;2\times {\left({B}_{p,r}&amp;#x0005E;{s-1}\left(\mathrm{\mathbb{R}}\right)\right)}&amp;#x0005E;2\left(s&gt;\max \left\{5/2,2&amp;#x0002B;1/p\right\},1\le p,r\le \infty \right) $$ are established by constructing new approximate solutions and initial data. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new 2‐microlocal Besov and Triebel–Lizorkin spaces of Jaffard type, which contain many classical functions. We establish local theory of these function spaces by the φ‐transform and the wavelet decomposition coefficients. Moreover, we give a characterization of these local spaces by a local polynomial approximation. [ABSTRACT FROM AUTHOR]

In the current study, we define Besov‐type spaces related to the Lebedev‐Skalskaya (LS‐) transform. We obtain Parseval's relation for continuous LS‐wavelet transform in L2(ℝ+;dx)$$ {L}^2\left({\mathrm{\mathbb{R}}}^{+}; dx\right) $$ space, and then using it, we extend the notion of continuous LS‐wavelet transform on Lp(ℝ+;dx)$$ {L}^p\left({\mathrm{\mathbb{R}}}^{+}; dx\right) $$ space and, in the end, derive continuity of LS‐wavelet transform on Besov‐type spaces and characterize the Besov LS‐space using LS‐wavelet coefficients. [ABSTRACT FROM AUTHOR]

The research presented in this paper is a continuation of the recent studies of new classes of integral operators on the unit disc 픻, which are called Hausdorff–Berezin operators. We consider general operators of Hausdorff–Berezin type constructed with an arbitrary positive Radon measure on the unit disc within the framework of weighted spaces with the so‐called Möbius weights. This is a fairly wide class of weights in the unit disc which includes classical radial weights with singularities or zeroes at the boundary. Sufficient conditions for the boundedness of the Hausdorff–Berezin operators in these spaces are obtained. In the case of nonnegative kernels, a boundedness criterion is obtained in Lp(픻). Approximation properties of operators of Hausdorff–Berezin type are also studied. We also discuss some applications. [ABSTRACT FROM AUTHOR]

This article develops an approach to unique, strong periodic solutions to quasilinear evolution equations by means of the classical Da Prato–Grisvard theorem on maximal Lp$L^p$‐regularity in real interpolation spaces. The method is used to show that quasilinear Keller–Segel systems admit a unique, strong T$T$‐periodic solution in a neighborhood of 0 provided the external forces are T$T$‐periodic and satisfy certain smallness conditions. A similar assertion applies to a Nernst–Planck–Poisson type system in electrochemistry. The proof for the quasilinear Keller–Segel systems relies also on a new mixed derivative theorem in real interpolation spaces, that is, Besov spaces, which is of independent interest. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of the Navier–Stokes equations in Rn$\mathbb {R}^n$ (n≥3)$n\ge 3)$ and establish several new extension criteria involving the pressure or its gradient. In particular, we improve the previous results by means of the homogeneous Besov space with negative differential orders in the case n=3$n=3$. Our method is based on the interpolation inequality and the trilinear estimate due to Gérard–Meyer–Oru (Séminaire É. D. P. (1996–1997), Exp. No. IV, 1–8) and Guo–Kučera–Skalák (J. Math. Anal. Appl. 458 (2018) 755–766), respectively. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the local higher fractional differentiability regularity estimates in Besov spaces of weak solutions for the quasilinear parabolic equations with general growth under some proper conditions on a,A$a, A$, and F$\mathbf {F}$. Moreover, we would like to point out that the results in the present work improve the known results for such parabolic equations. [ABSTRACT FROM AUTHOR]

We consider the well‐posedness of the second‐order degenerate differential equations (Mu′)′(t)+Λu′(t)+∫−∞ta(t−s)u′(s)ds=Au(t)+∫−∞tb(t−s)Bu(s)ds+f(t)(P)$$\begin{eqnarray*} &&\hspace*{18pc} (Mu^{\prime })^{\prime }(t) +\Lambda u^{\prime }(t)+\int _{-\infty }^t a(t-s)u^{\prime }(s)ds\\ &&\hspace*{18pc}\quad = Au(t)+ \int _{-\infty }^t b(t-s) Bu(s) ds + f(t)\qquad\qquad\qquad \mathrm{{(P)}} \end{eqnarray*}$$with infinite delay on [0, 2π] in Lebesgue–Bochner spaces Lp(T;X)$L^p(\mathbb {T}; X)$ and periodic Besov spaces Bp,qs(T;X)$B_{p,q}^s (\mathbb {T}; X)$, where A,B,Λ$A,\ B,\ \Lambda$, and M are closed linear operators in a Banach space X satisfying D(A)∩D(B)⊂D(M)∩D(Λ)$D(A)\cap D(B)\subset D(M) \cap D(\Lambda)$ and the kernels a,b∈L1(R+)$ a, b\in L^1(\mathbb {R}_+)$. Using known operator‐valued Fourier multiplier theorems, we are able to give necessary and sufficient conditions for the well‐posedness of (P) in Lp(T;X)$L^p(\mathbb {T}; X)$ and Bp,qs$B_{p,q}^s$(T;X)$(\mathbb {T}; X)$. These results are applied to examine some concrete examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of the d‐dimensional magneto‐micropolar equations (d=2$$ d=2 $$ or d=3$$ d=3 $$) 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. [ABSTRACT FROM AUTHOR]

We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space Rn$\mathbb {R}^n$ for n≥3$n \ge 3$. It is clarified that if w is small in Ḃp*,q′−1+np*$\dot{B}^{-1+\frac{n}{p_\ast }}_{p_\ast , q^{\prime }}$ for 1≤p*0$s > 0$. [ABSTRACT FROM AUTHOR]

In this paper, we consider traces at initial times for functions with mixed time‐space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equations, where uniform trace estimates on the half‐line are shown. [ABSTRACT FROM AUTHOR]

We 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. [ABSTRACT FROM AUTHOR]

We consider the stationary Navier–Stokes equations in the two‐dimensional torus T2$\mathbb {T}^2$. For any ε>0$\varepsilon >0$, we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces Ḃp+ε,q−1+2p(T2)$\dot{B}^{-1+\frac{2}{p}}_{p+\varepsilon , q}(\mathbb {T}^2)$ for given small external forces in Ḃp+ε,q−3+2p(T2)$\dot{B}^{-3+\frac{2}{p}}_{p+\varepsilon , q}(\mathbb {T}^2)$ when 1≤p<2$1\le p <2$. 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 (p,q)∈({2}×(2,∞])∪((2,∞]×[1,∞])$(p,q)\in (\lbrace 2\rbrace \times (2,\infty ])\cup ((2,\infty ]\times [1,\infty ])$, 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 Ḃp,q−3+2p(T2)$\dot{B}^{-3+\frac{2}{p}}_{p ,q}(\mathbb {T}^2)$ to Ḃp,q−1+2p(T2)$\dot{B}^{-1+\frac{2}{p}}_{p, q}(\mathbb {T}^2)$. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the regularity criteria of axisymmetric weak solutions to the three‐dimensional (3D) incompressible magnetohydrodynamics (MHD) equations with nonzero swirl component. By making use of techniques of the Littlewood–Paley decomposition, we show that weak solutions to the 3D axisymmetric MHD equations become regular if the swirl component of vorticity satisfies that wθeθ∈L1(0,T;Ḃ∞,∞0)$w_{\theta }e_{\theta }\in L^{1}\big (0,T;\dot{B}_{\infty ,\infty }^{0}\big)$, which partially gives a positive answer to the marginal case for the regularity of MHD equations. [ABSTRACT FROM AUTHOR]

The global existence of solutions to a damped MHD system in Besov spaces is obtained. This research is motivated by the open question [Wu et al., J. Nonlinear Sci. 25 (2015), 157–192, Remark 5.1] on the possibility of extension to the existence of global small solutions from a potential Banach space to a Besov space. The present result positively confirms the open question. What is more, the initial data are not required to be uniformly small. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D inhomogeneous incompressible magnetohydrodynamic (MHD) system. By the assumption of the smallness of initial velocity and magnetic fluids in the critical Besov space, the local and global well‐posedness of 3D inhomogeneous incompressible equations is obtained. It improves some previous results of MHD equations by generalizing the range of exponent p$$ p $$ in Besov spaces B˙p,13/p−1$$ {\dot{B}}_{p,1}^{3/p-1} $$ with 1

We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner‐type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species that is small enough. Moreover, the large solution is obtained for initial densities belonging to the low regularity Besov spaces with different regularity and integral indices, which indicates more specific coupling relations between the difference and the summation of negatively and positively charged densities. [ABSTRACT FROM AUTHOR]

In this paper, we make the first attempt to investigate the Cauchy problem of a shallow water model, namely, the inviscid lake equations, in the Besov spaces. Notably, we prove the global existence and uniqueness of the solutions in the Besov spaces Bp,qs(ℝ2)$$ {B}_{p,q}&amp;#x0005E;s\left({\mathbb{R}}&amp;#x0005E;2\right) $$ for s>2p+1$$ s&gt;\frac{2}{p}&amp;#x0002B;1 $$ and s=2p+1$$ s&amp;#x0003D;\frac{2}{p}&amp;#x0002B;1 $$ if q=1$$ q&amp;#x0003D;1 $$, which contain the particular case of the endpoint Besov space B∞,11(ℝ2)$$ {B}_{\infty, 1}&amp;#x0005E;1\left({\mathbb{R}}&amp;#x0005E;2\right) $$. [ABSTRACT FROM AUTHOR]

We study traces of weighted Triebel–Lizorkin spaces Fp,qs(Rn,w)$F^s_{p,q}(\mathbb {R}^n,w)$ on hyperplanes Rn−k$\mathbb {R}^{n-k}$, where the weight is of Muckenhoupt type. We concentrate on the example weight wα(x)=|xn|α$w_\alpha (x) = {\big\vert x_n\big\vert }^\alpha$ when |xn|≤1$\big\vert x_n\big\vert \le 1$, x∈Rn$x\in \mathbb {R}^n$, and wα(x)=1$w_\alpha (x)=1$ otherwise, where α>−1$\alpha >-1$. Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space (Bp1,p1s1(Rn−k),Bp2,p2s2(Rn−k))θ,r$\big (B^{s_1}_{p_1,p_1}\big (\mathbb {R}^{n-k}\big), B^{s_2}_{p_2,p_2}{\big (\mathbb {R}^{n-k}\big)\big)}_{\theta ,r}$, 00$s>0$ sufficiently large, 0<θ<1$0<\theta <1$, 0

We study multiplication as well as Nemytskij operators in anisotropic vector‐valued Besov spaces Bps,ω$B^{s, \omega }_p$, Bessel potential spaces Hps,ω$H^{s, \omega }_p$, and Sobolev–Slobodeckij spaces Wps,ω$W^{s, \omega }_p$. Concerning multiplication we obtain optimal estimates, which constitute generalizations and improvements of known estimates in the isotropic/scalar‐valued case. Concerning Nemytskij operators we consider the acting of analytic functions on supercritial anisotropic vector‐valued function spaces of the above type. Moreover, we show how the given estimates may be used in order to improve results on quasilinear evolution equations as well as their proofs. [ABSTRACT FROM AUTHOR]

Considered 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 Hs(R)$H^{s}(\mathbf {R})$ with s>9/2$s>9/2$ 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 B2,rs(T)(s>9/2,1≤r≤∞)$B^{s}_{2,r}(\mathbf {T})\, (s>9/2, 1\le r\le \infty)$ and B2,19/2(T)$B^{9/2}_{2,1}(\mathbf {T})$ are proved. Finally, the persistence property of solutions for this problem is studied. [ABSTRACT FROM AUTHOR]

One of the most important models for understanding chemotactic aggregation is the Keller–Segel system. We analyze the time‐fractional Keller–Segel system in a new framework; we derive the global well‐posedness and the asymptotic behavior of solutions in critical Besov‐weak‐Herz spaces that consist in Besov spaces based on weak‐Herz spaces. [ABSTRACT FROM AUTHOR]

We use Bochner's subordination technique to obtain caloric smoothing estimates in Besov‐ and Triebel–Lizorkin spaces. Our new estimates extend known smoothing results for the Gauß–Weierstraß, Cauchy–Poisson and higher‐order generalized Gauß–Weierstraß semigroups. Extensions to other function spaces (homogeneous, hybrid) and more general semigroups are sketched. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general nonparametric regression estimation model with the feature of having multiplicative noise. We propose a linear estimator and nonlinear estimator by wavelet method. The convergence rates of those regression estimators under pointwise error over Besov spaces are proved. It turns out that the obtained convergence rates are consistent with the optimal convergence rate of pointwise nonparametric functional estimation. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the optimal time decay rates of classical solutions to the 3D compressible Navier-Stokes equations. Based on a global priori estimate, the optimal time decay rates of the solution and its first order derivative in L2-norm are obtained if H l -norm (l ≥ 4) of the initial perturbation around a constant state is small enough and ∥ (ρ 0 − ρ ¯, m 0) ∥ B ̇ 1, ∞ − s ℓ is bounded for s ∈ [ 0, 1 2). Compared with the previous works by Hai-Liang Li and Ting Zhang (Math Meth Appl Sci 34:670-682, 2011), we remove the smallness of ∥ (ρ 0 − ρ ¯, m 0) ∥ B ̇ 1, ∞ − s and our condition involves only the low frequencies of the data in B ̇ 1, ∞ − s (R 3). [ABSTRACT FROM AUTHOR]

The present paper is dedicated to the large‐time asymptotic behavior of global strong solutions near constant equilibrium (away from vacuum) to the compressible Navier‐Stokes‐Poisson equations in the Lp critical framework. The proof mainly depends on the pure energy argument without the spectral analysis (for the linearized system), which enables us to remove the usual smallness assumption on low frequencies of initial data. It is improved in comparison with the result of Shi and Xu [J Differ Equ. 2019; 266(10):6426‐6458]. [ABSTRACT FROM AUTHOR]

Regularity criteria of the weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamic (MHD) equations are discussed. Our results imply that the scalar pressure field π plays an important role in the regularity problem of MHD equations. We derive that the weak solution u , b is regular on 0 , T , which is provided for the scalar pressure field π in the Besov spaces. [ABSTRACT FROM AUTHOR]