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In this paper we study the harmonic map heat flow on the euclidean space R d and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B ˙ p , ∞ d p (R d) where d < p < ∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖ ∇ u (t) ‖ L ∞ (R d) ≤ C t − 1 2 . Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity. [ABSTRACT FROM AUTHOR]

In the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa–Holm equation on the real-line case. We show that the data-to-solution map of the Camassa–Holm equation is not uniformly continuous on the initial data in Besov spaces B p , r s (R) with s > 1 2 and 1 ≤ p , r < ∞ , which improves the previous works Himonas et al. (2007) [23] , Li et al. (2020) [32] and Li et al. (2021) [31]. Furthermore, we present a strengthening of our previous work in Li et al. (2020) [32] and prove that the data-to-solution map for the Camassa–Holm equation is nowhere uniformly continuous in B p , r s (R) with s > max ⁡ { 1 + 1 / p , 3 / 2 } and (p , r) ∈ [ 1 , ∞ ] × [ 1 , ∞). The method applies also to the b-family of equations which contain the Camassa–Holm and Degasperis–Procesi equations. [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]

In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established 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, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces B ˙ p , 1 2 p (R 2) × B ˙ p , q 2 p − 1 (R 2). As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in B ˙ p , 1 2 p (R 2) × B ˙ p , 1 2 p − 1 (R 2) for p < 4 and ill-posed in B ˙ p , 1 2 p (R 2) × B ˙ p , 1 2 p − 1 (R 2) for p > 4. In this paper, we prove that this system is ill-posed for the critical case p = 4 in the sense of "norm inflation". Furthermore, we also show that the system is ill-posed in B ˙ 4 , 1 1 2 (R 2) × B ˙ 4 , q − 1 2 (R 2) for any q ≠ 2. [ABSTRACT FROM AUTHOR]

Let Δ d be the discrete Laplacian defined on Z d by setting Δ d f (n →) = ∑ j = 1 d − [ f (n → + e → j) + f (n → − e → j) − 2 f (n →) ] , n → ∈ Z d , where { e → j : j = 1 , ... , d } is the standard basis for R d. In this paper, we prove weighted mixed norm estimates and end-point estimates for the maximal regularity of the discrete parabolic equation { u t + Δ d u = f , t ∈ [ 0 , T) u (0 , ⋅) = 0 , where T ∈ (0 , ∞). [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Cauchy problem for the Fornberg–Whitham equation is not locally well-posed in B p , r s (R) with (s , p , r) ∈ (1 , 1 + 1 p) × [ 2 , ∞) × [ 1 , ∞ ] or (s , p , r) ∈ { 1 } × [ 2 , ∞) × [ 1 , 2 ] by showing norm inflation phenomena of the solution for some special initial data. [ABSTRACT FROM AUTHOR]

For the famous Camassa-Holm equation, the well-posedness in C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 1 ≤ p ≤ 2 and the ill-posedness in C ([ 0 , T ] ; B ∞ , 1 1 (R)) had been studied in [15,16,23]. That is to say, it left an open problem in the critical case C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 2 < p < + ∞ proposed by Danchin in [15,16]. In this paper, we solve this problem and obtain the local well-posedness for the Camassa-Holm equation in critical Besov spaces C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 1 ≤ p < + ∞. It is worth mentioning that our method is suitable for many Camassa-Holm type equations, such as the Novikov equation and the two-component Camassa-Holm system, and can also improve their index of the local well-posedness. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem for a two-component Camassa-Holm system with high order nonlinearity, which is a multi-component extension of the Fokas-Olver-Rosenau-Qiao equation. Firstly, we state the local well-posedness for the Cauchy problem of the system in the framework of Sobolev-Besov spaces. Then, we establish the precise blow-up mechanism for the strong solutions by means of the transport equation theory. As is well-known, the H 1 -norm conservation law of the velocity component is crucial to study blow-up phenomena of the single Camassa-Holm type equation. However, it is no longer available for our nonlinear coupled system. To overcome this difficulty, we derive some suitable conservation laws by sufficiently exploiting the fine structure of the system. Based on which, we finally construct several new blow-up strong solutions with certain initial profiles in finite time. [ABSTRACT FROM AUTHOR]

This paper is concerned with the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation. For the generalized singular integral operator, we obtain uniform L p − L q estimates with respect to a parameter β. From this one can cover the L p -boundedness of the Calderón-Zygmund operator with rough kernel by letting β → 0. We applied this estimate to study the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation. Local well-posedness in the Besov space B p , q s and some limit behaviour of the solutions are obtained. Our results improve the previous ones by Yu-Zheng-Jiu in 2019 and by Yu-Jiu-Li in 2021. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the higher order fractional differentiability of weak solution pairs for the following system modeling the stationary motion of electro-rheological fluids { − div ((1 + | D u | 2) p (x) − 2 2 D u) + [ ∇ u ] u + ∇ π = − div F in Ω div u = 0 in Ω in Besov space, where F is Besov regular and the non-standard growth exponent p (x) is Besov-Orlicz regular. The main approach to prove this result is to establish a Besov regular priori estimate of weak solution pairs for its Stokes setting without convective term [ ∇ u ] u , and then treat the convective term as a part of nonhomogeneous term in the general systems. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the Cauchy problem of the cubic Camassa-Holm-type equations which arise as an asymptotic model with a nonlocal cubic nonlinearity for the unidirectional propagation of shallow water waves, which admit the single peaked solitons and multi-peakon solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model. Then, by overcoming the difficulties cased by the complicated mixed nonlinear structure, we present a precise blow-up scenario. Moreover, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions for Cauchy problem in the Gevrey-Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. Finally, we investigate the persistence properties of the solution to the Cauchy problem, we prove that the solution maintains the corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we first show by constructing a special initial data that the solution map for the one dimensional hyperbolic Keller-Segel equations (HKSE) starting from u 0 is discontinuous at t = 0 in the metric of B 2 , ∞ s (R) , s > 3 2. Then, we establish the Hadamard local well-posedness result for the high dimensional HKSE in the larger Besov spaces B p , 1 1 + d p (R d) , 1 ≤ p < ∞ , which improves the local theory proved by [Zhou, Zhang & Mu, J. Differ. Equ. , 302(2021), pp.662-679]. Moreover, we investigate the inviscid limit of the Keller-Segel equations with small diffusivity ϵ Δ u as ϵ → 0 in the same topology of Besov spaces as the initial data. Finally, we establish two kinds of blow-up criteria for strong solutions in Besov spaces by means of the Littlewood-Paley theory. [ABSTRACT FROM AUTHOR]

We are interested in the multi-dimensional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the "div-curl" structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. In absence of compatible conditions, the new effective flux is introduced, which enables us to capture the dissipation arising from combination of density and deformation tensor. The partial dissipation in non-Newtonian compressible fluids, is weaker than that of classical Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

For the famous Camassa-Holm equation, the well-posedness in B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞) and the ill-posedness in B p , r 1 + 1 p (R) with p ∈ [ 1 , ∞ ] , r ∈ (1 , ∞ ] had been studied in [13,14,16,23] , that is to say, it only left an open problem in the critical case B ∞ , 1 1 (R) proposed by Danchin in [13,14]. In this paper, we solve this problem by proving the norm inflation and hence the ill-posedness for the Camassa-Holm equation in B ∞ , 1 1 (R). Therefore, the well-posedness and ill-posedness for the Camassa-Holm equation in all critical Besov spaces B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞ ] have been completed. Finally, since the norm inflation occurs by choosing an special initial data u 0 ∈ B ∞ , 1 1 (R) but u 0 x 2 ∉ B ∞ , 1 0 (R) (an example implies B ∞ , 1 0 (R) is not a Banach algebra), we then prove that this condition is necessary. That is, if u 0 x 2 ∈ B ∞ , 1 0 (R) holds, then the Camassa-Holm equation has a unique solution u (t , x) ∈ C T (B ∞ , 1 1 (R)) ∩ C T 1 (B ∞ , 1 0 (R)) and the norm inflation will not occur. [ABSTRACT FROM AUTHOR]

In this paper, we prove the global Hadamard well-posedness of strong solutions to a non-isospectral two-component cubic Camassa-Holm system in the critical Besov space B 2 , 1 1 / 2 (T). Our results show that in comparison with the well-known work for classic Camassa-Holm-type equations, the existence of global solution only relies on the L 1 -integrability of the variable coefficients α (t) and γ (t) , but nothing to do with the shape of the initial data. The key ingredient of the proof hinges on the careful analysis of the mutual effect among two component forms, the uniform bound of approximate solutions, and several crucial estimates of cubic nonlinearities in low-regularity Besov spaces via the Littlewood-Paley decomposition theory. A reduced case in our results yields the global existence of solutions in a Besov space for two kinds of well-known isospectral peakon system with weakly dissipative terms. Moreover, we derive two kinds of precise blow-up criteria for a strong solution in both critical and non-critical Besov spaces, as well as providing specific characterization for the lower bound of the blow-up time, which implies the global existence with additional conditions on the time-dependent parameters α (t) an γ (t). We believe that the method in this paper can be applied to the study of well-posedness for other transport type equations. [ABSTRACT FROM AUTHOR]

This paper presents a new small data global well-posedness result on the incompressible Oldroyd-B model with only dissipation in the equation of stress tensor (without stress tensor damping or velocity dissipation). The dissipation is not necessarily given by the standard Laplacian operator and any fractional dissipation with fractional power equal to or greater than 1/2 suffices. The functional setting is the hybrid homogeneous Besov spaces, which allow us to maximize the functional spaces of the initial data. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the Cauchy problem of the Doi-Edwards polymer model with dimension d ≥ 2. The model was derived in the late 1970s to describe the dynamics of polymers in melts. The system contains a Navier-Stokes equation with an additional stress tensor which depend on the deformation gradient tensor and the memory function. The deformation gradient tensor satisfies a transport equation and the memory function satisfies a degenerate parabolic equation. We first proved the local well-posedness for the Doi-Edwards polymer model in Besov spaces by using the Littlewood-Paley theory. Moreover, if the initial velocity and the initial memory is small enough, we obtain a global existence result. [ABSTRACT FROM AUTHOR]

In this paper, we give a construction of u 0 ∈ B p , ∞ σ such that the corresponding solution to the Camassa-Holm equation starting from u 0 is discontinuous at t = 0 in the metric of B p , ∞ σ , which implies the ill-posedness for this equation in B p , ∞ σ. We also apply our method to the b-equation and Novikov equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D inhomogeneous incompressible asymmetric fluids system with density-dependent viscosity. By the assumption of the smallness of initial velocity in the critical Besov space, B ˙ p , 1 3 / p − 1 for 1 < p < 6 and the initial density in the critical Besov space and bounded away from vacuum, the local and global well-posedness of 3D inhomogeneous incompressible asymmetric fluids is obtained. By giving the different estimates for pressure in the system for 1 < p < 3 2 and 3 2 ≤ p < 6 , it shows the a priori estimate for the corresponding linearized equation. This not only improves the previous results for 3D inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity, but also obtains the new results on the micropolar system without smallness for density in critical Besov spaces. [ABSTRACT FROM AUTHOR]

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space (X , d X , μ X) satisfying a 2-Poincaré inequality. Given a bounded domain Ω ⊂ X with μ X (X ∖ Ω) > 0 , and a function f in the Besov class B 2 , 2 θ (X) ∩ L 2 (X) , we study the problem of finding a function u ∈ B 2 , 2 θ (X) such that u = f in X ∖ Ω and E θ (u , u) ≤ E θ (h , h) whenever h ∈ B 2 , 2 θ (X) with h = f in X ∖ Ω. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on Ω, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extends the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the inviscid limit for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. By virtue of the Littlewood-Paley theory, we first obtain a uniform estimate for the solution to the FENE dumbbell model with viscosity in Besov spaces. Moreover, we show that the data-to-solution map is continuous. Finally, we prove that the strong solution of the FENE dumbbell model converges to a Euler system couple with a Fokker-Planck equation. Furthermore, convergence rates in Lebesgue spaces are obtained also. [ABSTRACT FROM AUTHOR]

In this paper, we first study the local well-posedness for the Cauchy problem of the hyperbolic Keller-Segel equation in Besov spaces B p , r s (R d) with 1 ≤ p , r ≤ + ∞ and s > 1 + d p , i.e., the local existence, unique and continuous dependence on the initial data for the solution of this system are obtained, then we further show that this data-to-solution map is not uniformly continuous in these Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrödinger equation with spatial inhomogeneity coefficient K (x) behaves like | x | − b for 0 < b < min ⁡ { N 2 , 4 }. We show the local well-posedness in the whole H s -subcritical case, with 0 < s ≤ 2. The difficulties of this problem come from the singularity of K (x) and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrödinger equations in Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global existence and time decay rates of the solutions to Boltzmann equation for hard potential in the critical Chemin-Lerner type space. The evolution of the energy in negative Besov space is derived to be preserved. Our results are based on the modified trilinear estimates and some new observations. We use the Lyapunov-type inequality of the energy functionals to show the time decay rates without linear decay analysis. Compared with the previous work, the decay rates of the global solution are established for lower-regularity initial data. [ABSTRACT FROM AUTHOR]

Let H = − Δ + | x | 2 be the harmonic oscillator on R n. In this paper, we prove estimates on Besov spaces associated to the operator H and the end-point maximal regularity estimates for the fractional harmonic oscillator H α , 0 < α ≤ 1 , on the Besov spaces associated to the harmonic oscillator H. These spaces are the appropriate function spaces for the study of estimates on Besov type spaces and the end-point maximal regularity estimates for the fractional power H α in the sense that similar estimates might fail with the classical Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study some conditions related to the question of the possible blow-up of regular solutions to the 3D Navier-Stokes equations. In particular, up to a modification in a proof of a very recent result from [1] , we prove that if one component of the velocity remains small enough in a sub-space of H ˙ 1 2 "almost" scaling invariant, then the 3D Navier-Stokes equations are globally wellposed. In a second time, we investigate the same question under some conditions on one component of the vorticity and unidirectional derivative of one component of the velocity in some critical Besov spaces of the form L T p ( B ˙ 2 , ∞ α , 2 p − 1 2 − α) or L T p ( B ˙ q , ∞ 2 p + 3 q − 2). [ABSTRACT FROM AUTHOR]

The present paper is dedicated to the Cauchy problem of the two dimensional chemotaxis-shallow water system in the critical Besov spaces. We first obtain the global solutions to this system with small initial data and then investigate the optimal time decay rate of the solutions by a suitable energy argument (independent of the spectral analysis). Moreover, the bacterial density is allowed to vanish here. [ABSTRACT FROM AUTHOR]

In the paper, we consider the initial value problem to the Camassa-Holm equation in the real-line case. Based on the local well-posedness result and the lifespan, we proved that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces in the sense of Hadamard. This result improves considerably the previous work given by Himonas–Kenig [23]. [ABSTRACT FROM AUTHOR]

Considered in this paper is the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa-Holm equation (also called Fokas-Olver-Rosenau-Qiao equation) and the Novikov equation as two special cases. The main investigation is the Cauchy problem of the new shallow-water model with qualitative properties of its solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model, and then derive a precise blow-up scenario and the blow-up mechanisms for solutions with certain initial profiles. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we investigate the persistence properties of the solution to the Cauchy problem of the new shallow-water model. [ABSTRACT FROM AUTHOR]

In this paper, we establish analyticity of solutions to the barotropic compressible Navier-Stokes equations describing the motion of the density ρ and the velocity field u in R 3. We assume that ρ 0 is a small perturbation of 1 and (1 − 1 / ρ 0 , u 0) are analytic in Besov spaces with analyticity radius ω > 0. We show that the corresponding solutions are analytic globally in time when (1 − 1 / ρ 0 , u 0) are sufficiently small. To do this, we introduce the exponential operator e (ω − θ (t)) D acting on (1 − 1 / ρ , u) , where D is the differential operator whose Fourier symbol is given by | ξ | 1 = | ξ 1 | + | ξ 2 | + | ξ 3 | and θ (t) is chosen to satisfy θ (t) < ω globally in time. [ABSTRACT FROM AUTHOR]

Let L a be a Schrödinger operator with inverse square potential a | x | − 2 on R n , n ≥ 3. The main aim of this paper is to develop the theory of new Besov and Triebel–Lizorkin spaces associated to L a based on the new space of distributions. As applications, we apply the theory to study some problems on the parabolic equation associated to L a. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the initial value problem of the periodic Euler-Poincaré equations. We first present a new blow-up result to the system for a special class of smooth initial data by using the rotational invariant properties of the system. Then, we prove that the periodic Euler-Poincaré equations are ill-posed in critical Besov spaces by a contradiction argument. Finally, we verify the system possesses a class of peakon solutions in the sense of distributions. [ABSTRACT FROM AUTHOR]

In this paper, we study the periodic Cauchy problem for a two-component non-isospectral cubic Camassa-Holm system which includes the Fokas-Olver-Rosenau-Qiao (FORQ) or modified Camassa-Holm (MCH) equation and the two-component MCH system as two special cases. The system is integrable in the sense of possessing a non-isospectral Lax pair with the spectrum depending on time t , and admits multi-peakon solutions in an explicit form. Furthermore, we establish the local well-posedness for the system in the Besov space B 2 , r s (T) with s > 3 / 2 , 1 ≤ r ≤ ∞ , where the key ingredients include the Friedrichs regularization method, the Littlewood-Paley decomposition theory, and the transport theory in Besov spaces. Then we derive a precise blow-up criteria, which is dependent of the parameters α (t) and γ (t). Moreover, by the intrinsic structure of the system, we obtain a new blow-up result for strong solutions with sufficient conditions on the initial data and parameters. The entire proof procedure relies upon a newly derived transport equation which is involved in nonlocal velocity term along the characteristic curves. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an asymptotic model for wave propagation in shallow water with the effect of the Coriolis force. We first establish the local well-posedness in a range of the Besov spaces, as well as the local well-posedness in the critical space. Then, we study the Gevrey regularity of the shallow-water model by using a generalized Cauchy-Kovalevsky theorem, which implies that the shallow-water model admits analytical solutions locally in time and globally in space. Moreover, we obtain a precise lower bound of the lifespan and the continuity of the solution. Finally, working with moderate weight functions that are commonly used in time-frequency analysis, some persistence results to the shallow-water model are illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear evolution equation modelling the propagation of surface waves in the shallow water regime of large amplitude, which is characterised by some cubical nonlinearities. First, we establish the local well-posedness in Besov space B 2 , 1 3 / 2. Then, we give a blow-up criterion. Finally, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time. [ABSTRACT FROM AUTHOR]

Abstract This paper addresses a well-posedness of the weak solution to the transport equation (describing how a scalar quantity is transported in a space) with an initial data in the scaling invariant Besov or Essén–Janson–Peng–Xiao space via the boundedness of the left and right compositions acting on each space. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces H s , s > 3 / 2 for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in H r -topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. [ABSTRACT FROM AUTHOR]

Abstract The compressible Navier–Stokes–Poisson system takes the form of usual Navier–Stokes equations coupled with the self-consistent Poisson equation, which is used to simulate the transport of charged particles under the electrostatic potential force. In this paper, we focus on the large-time behavior of global strong solutions in the L p critical Besov spaces. Inspired by the dissipative effect arising from Poisson potential, we formulate a new regularity assumption of low frequencies and then establish the sharp time-weighted inequality, which leads to the optimal time-decay estimates of strong solutions. Indeed, we see that the decay of density is faster at the half rate than that of velocity, which is a different ingredient in comparison with the situation of compressible Navier–Stokes equations. Our proof mainly depends on tricky and non classical Besov product estimates with respect to various Sobolev embeddings. [ABSTRACT FROM AUTHOR]

In recent paper [7] , Y. Du and K. Wang (2013) proved that the global-in-time Koch–Tataru type solution ( u , d ) to the n -dimensional incompressible nematic liquid crystal flow with small initial data ( u 0 , d 0 ) in BMO − 1 × BMO has arbitrary space–time derivative estimates in the so-called Koch–Tataru space norms. The purpose of this paper is to show that the Koch–Tataru type solution satisfies the decay estimates for any space–time derivative involving some borderline Besov space norms. More precisely, for the global-in-time Koch–Tataru type solution ( u , d ) to the nematic liquid crystal flow with initial data ( u 0 , d 0 ) ∈ BMO − 1 × BMO and ‖ u 0 ‖ BMO − 1 + [ d 0 ] BMO ≤ ε for some small enough ε > 0 , and for any positive integers k and m , one has ‖ t k + m 2 ( ∂ t k ∇ m u , ∂ t k ∇ m ∇ d ) ‖ L ˜ ∞ ( R + , B ˙ ∞ , ∞ − 1 ) ∩ L ˜ 1 ( R + ; B ˙ ∞ , ∞ 1 ) ≤ ε . Furthermore, we shall give that the solution admits a unique trajectory which is Hölder continuous with respect to space variables. [ABSTRACT FROM AUTHOR]

For fractional Navier–Stokes equations and critical initial spaces X , one used to establish the well-posedness in the solution space which is contained in C ( R + , X ) . In this paper, for heat flow, we apply parameter Meyer wavelets to introduce Y spaces Y m , β where Y m , β is not contained in C ( R + , B ˙ ∞ 1 − 2 β , ∞ ) . Consequently, for 1 2 < β < 1 , we establish the global well-posedness of fractional Navier–Stokes equations with small initial data in all the critical oscillation spaces. The critical oscillation spaces may be any Besov–Morrey spaces ( B ˙ p , q γ 1 , γ 2 ( R n ) ) n or any Triebel–Lizorkin–Morrey spaces ( F ˙ p , q γ 1 , γ 2 ( R n ) ) n where 1 ≤ p , q ≤ ∞ , 0 ≤ γ 2 ≤ n p , γ 1 − γ 2 = 1 − 2 β . These critical spaces include many known spaces. For example, Besov spaces, Sobolev spaces, Bloch spaces, Q-spaces, Morrey spaces and Triebel–Lizorkin spaces etc. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the well-posedness for the 3-D inhomogeneous incompressible Navier–Stokes equations with variable viscosity. With some smallness assumption on the BMO-norm of the initial density, we first get the local well-posedness of (1.1) in the critical Besov spaces. Moreover, if the viscosity coefficient is a constant, we can extend this local solution to be a global one. Our theorem implies that we have successfully extended the integrability index p of the initial velocity which has been obtained by Abidi, Gui and Zhang in [3] , Burtea in [8] and Zhai and Yin in [32] to approach the ideal one i.e. 1 < p < 6 . The main novelty of this work is to apply the CRW theorem obtained by Coifman, Rochberg, Weiss in [11] to get a new a priori estimate for an elliptic equation with variable coefficients. The uniqueness of the solution also relies on a Lagrangian approach as in [16–18] . [ABSTRACT FROM AUTHOR]

In this paper, we consider the large time behavior of global solutions to the initial value problem for the compressible Navier–Stokes–Poisson system in the L p critical framework and in any dimension N ≥ 3 . We obtain the time decay rates, not only for Lebesgue spaces, but also for a family of Besov norms with negative or nonnegative regularity exponents , which improves the decay results in high Sobolev regularity. The proof is mainly based on the Littlewood–Paley theory and refined time weighted inequalities in Fourier space. [ABSTRACT FROM AUTHOR]

In this paper, we prove well-posedness of the Fornberg–Whitham equation in Besov spaces B 2 , r s in both the periodic and non-periodic cases. This will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data-to-solution map provided that the initial data belongs to B 2 , r s . We also establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from any bounded subset of B 2 , r s to C ( [ − T , T ] ; B 2 , r s ) . Furthermore, we prove a Cauchy–Kowalevski type theorem for this equation that establishes the existence and uniqueness of real analytic solutions and also provide blow-up criterion for solutions. [ABSTRACT FROM AUTHOR]

In Guo and Wang (2012) [10] , Y. Guo and Y. Wang developed a general new energy method for proving the optimal time decay rates of the solutions to dissipative equations. In this paper, we generalize this method in the framework of homogeneous Besov spaces. Moreover, we apply this method to a model arising from electro-hydrodynamics, which is a strongly coupled system of the Navier–Stokes equations and the Poisson–Nernst–Planck equations through charge transport and external forcing terms. We show that some weighted negative Besov norms of solutions are preserved along time evolution, and obtain the optimal time decay rates of the higher-order spatial derivatives of solutions by the Fourier splitting approach and the interpolation techniques. [ABSTRACT FROM AUTHOR]

In [18] Paicu and Zarnescu have studied an order tensor system which describes the flow of a liquid crystal. They have proven the existence of weak solutions, the propagation of higher regularity, namely H s with s > 1 and the weak-strong uniqueness in dimension two. This paper is devoted to fill the gap of their results, namely to propagate the low regularity, namely H s for 0 < s < 1 and to prove the uniqueness of the weak solutions. For the completeness of this research, we also propose an alternative approach in order to prove the existence of weak solutions. [ABSTRACT FROM AUTHOR]

The present paper is dedicated to the global well-posedness for the 3D inhomogeneous incompressible Navier–Stokes equations, in critical Besov spaces without smallness assumption on the variation of the density. We aim at extending the work by Abidi, Gui and Zhang (2012) [2] , and (2013) [3] to a lower regularity index about the initial velocity. The key to that improvement is a new a priori estimate for an elliptic equation with nonconstant coefficients in Besov spaces which have the same degree as L 2 in R 3 . Finally, we also generalize our well-posedness result to the inhomogeneous incompressible MHD equations. [ABSTRACT FROM AUTHOR]

In this paper, we prove the solution map of the Cauchy problem of Camassa–Holm type equations depends continuously on the initial data in nonhomogeneous Besov spaces in the sense of Hadamard by using the Littlewood–Paley theory and the method introduced by Kato [37] and Danchin [21] . [ABSTRACT FROM AUTHOR]

In this paper, we first prove the local well-posedness of the 2-D incompressible Navier–Stokes equations with variable viscosity in critical Besov spaces with negative regularity indices, without smallness assumption on the variation of the density. The key is to prove for p ∈ ( 1 , 4 ) and a ∈ B ˙ p , 1 2 p ( R 2 ) that the solution mapping H a : F ↦ ∇ Π to the 2-D elliptic equation div ( ( 1 + a ) ∇ Π ) = div F is bounded on B ˙ p , 1 2 p − 1 ( R 2 ) . More precisely, we prove that ‖ ∇ Π ‖ B ˙ p , 1 2 p − 1 ≤ C ( 1 + ‖ a ‖ B ˙ p , 1 2 p ) 2 ‖ F ‖ B ˙ p , 1 2 p − 1 . The proof of the uniqueness of solution to (1.2) relies on a Lagrangian approach [15–17] . When the viscosity coefficient μ ( ρ ) is a positive constant, we prove that (1.2) is globally well-posed. [ABSTRACT FROM AUTHOR]

In this paper, we obtain global existence and energy decay for 3D Hall-magnetohydrodynamics (Hall-MHD) system with − Δ u and − Δ B . Besides the classical energy method and Besov space techniques, the interpolating inequalities are crucial in the proof of decay estimates. Then two Osgood type blow-up criteria are established. Our results improve the corresponding theorems in [3] and [4] . In addition, we establish two Beale–Kato–Majda blow-up criterion for the generalized version of Hall-MHD with − Δ u and ( − Δ ) β B , β > 1 . [ABSTRACT FROM AUTHOR]