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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 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]

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 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 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]

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]

Abstract: Considered in this paper is the modified Camassa–Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse equation. The main investigation is the Cauchy problem of the modified Camassa–Holm equation with qualitative properties of its solutions. It is firstly shown that the equation is locally well-posed in a range of the Besov spaces. The blow-up scenario and the lower bound of the maximal time of existence are then determined. A blow-up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated. [Copyright &y& Elsevier]

Abstract: In this paper, we study a multidimensional bipolar hydrodynamic model for semiconductors or plasmas. This system takes the form of the bipolar Euler–Poisson model with electric field and frictional damping added to the momentum equations. In the framework of the Besov space theory, we establish the global existence of smooth solutions for Cauchy problems when the initial data are sufficiently close to the constant equilibrium. Next, based on the special structure of the nonlinear system, we also show the uniform estimate of solutions with respect to the relaxation time by the high- and low-frequency decomposition methods. Finally we discuss the relaxation-time limit by compact arguments. That is, it is shown that the scaled classical solution strongly converges towards that of the corresponding bipolar drift-diffusion model, as the relaxation time tends to zero. [Copyright &y& Elsevier]