In this paper, we are concentrated on demonstrating the Liouville type theorem for the stationary incompressible Magnetohydrodynamic equations in the whole space, the half-space, or a periodic slab, and presenting the solution must vanish under the condition that for some 0 ≤ δ ≤ 1 < L and q = 6(3 - δ)6 - δ,lim inf_R →∞1 R‖(u,h)‖_R < | x | < LR^3 - δ = 0 . We also deduce sufficient conditions by allowing shrinking ratio L = 1 + R−α. When in slab with zero boundary condition, stronger decay rate is needed. We do not assume the global bound of the velocity field u and the magnetic field h and investigate the Liouville type theorems by the conditions lim inf rather than lim.

We study the Hausdorff dimension of divergence sets for the convergence rate of fractional Schrödinger operators e^it(-Δ)^m 2 f , where f ∈ Hs. All results are sharp except at the endpoints.

In this paper, we establish the almost everywhere convergence of solutions to the Schrödinger operator with complex time P_γf(x,t) in higher dimensions, under the assumption that the initial data f belongs to the Sobolev space H^s(ℝ^d).

We analyze convergence rate of Schrödinger operator along curves U_γf(x,t), where f ∈ H^s(ℝ^d). All results are sharp up to the endpoints.

For Schr & ouml;dinger-type operators in one dimension, we consider the relationship between the convergence rate and the regularity for initial data. By establishing the associated frequency-localized maximal estimates, we prove sharp results up to the endpoints. The optimal range for the wave operator in all dimensions is also obtained.

Non-alcoholic fatty liver disease (NAFLD) is rapidly emerging as a leading etiology of chronic liver disease (CLD) in Asia. The increasing incidence of NAFLD is projected to drive a surge in NAFLD-related hepatocellular car-cinoma (HCC). A notable characteristic of NAFLD-HCC is its capacity for development in individuals without cirrhosis in more than a third of patients. Most practice guidelines recommend biannual ultrasound screening for patients with cirrhosis. In cases of severe limitations to ultrasound visualisation, cross-sectional abdominal imaging may be warranted. Improved strategies for HCC risk stratification are required for people with NAFLD but without cirrhosis. In this Review, we discuss the evolving trends of NAFLD and HCC in Asia, and implica-tions for surveillance. ( J CLIN EXP HEPATOL 2024;14:101213)

Note that some classic fluid dynamical systems such as the Navier-Stokes equations, Magnetohydrodynamics (MHD for short), Boussinesq equations and etc., are observably different from each other but obey some energy inequalities of the similar type. In this paper, the authors attempt to axiomatize the extending mechanism of solutions to these systems, merely starting from several basic axiomatized conditions such as the local existence, joint property of solutions and some energy inequalities. The results established have nothing to do with the concrete forms of the systems and, thus, give the extending mechanisms in a unified way to all systems obeying the axiomatized conditions. The key tools are several new multiplicative interpolation inequalities of Besov type, which have their own interests.

In the paper, we study derivative estimates of the iterated spherical averages (At)N(f). We obtain the optimal range of exponents (α, N, p) to ensure the Lp boundedness of $$P\left( {{\partial \over {\partial x}}} \right){\left( {{A_1}} \right)^N}(f)$$ for 1 ≤ p ≤ ∞, where P is a homogeneous polynomial of degree α. The main theorem extends some known results. As an application, we obtain the smallest N such that (A1)N: Lp(ℝn) → L (ℝn).

For the Cauchy problem to Keller-Segel system, we show well-posedness and time-decay estimates in the critical scaling-invariant Besov spaces by using Littlewood-Paley analysis together with the decay estimates of heat kernels.

In [3], Chen, Deng, Ding and Fan proved that the fractional power dissipative operator is bounded on Lebesgue spaces L p (ℝ n ), Hardy spaces H p (ℝ n ) and general mixed norm spaces, which implies almost everywhere convergence of such operator. In this paper, we study the rate of convergence on fractional power dissipative operator on some sobolev type spaces.

Almost everywhere convergence on the solution of Schrodinger equation is an important problem raised by Carleson in harmonic analysis. In recent years, this problem was essentially solved by building the sharp L-p-estimate of Schrodinger maximal function. Du-Guth-Li in [8] proved the sharp L-p-estimates for all p >= 2 in R2+1. Du-Zhang in [12] proved the sharp L-2-estimate in Rn+1 with n = 3, but for p > 2 the sharp Lpestimate of Schrodinger maximal function is still unknown. In this paper, we obtain partial results on this problem by using polynomial partitioning and refined Strichartz estimates. (C) 2021 Elsevier Inc. All rights reserved.

In this paper, we focused on demonstrating the Liouville type theorem for the stationary magnetohydrodynamic equations. First, we improve the well-known L92(R3) result logarithmically. In addition, we are aimed to an asymptotic formula for an integral involving the total head pressure Q=12|u|2+|h|2+p and its derivative over domains enclosed by level surfaces of Q, which provides us with the additional sufficient condition for the triviality of solution for the magnetohydrodynamic equations.

In this paper, we are concentrated on demonstrating the Liouville type theorem for the stationary Magnetohydrodynamic equations in mixed-norm Lebesgue spaces and weighted mixed-norm Lebesgue spaces. In particular, we show that, under some sufficient conditions in (weighted) mixed-norm Lebesgue spaces, the solution of stationary MHDs are identically zero. Precisely, we investigate solutions of MHDs that may decay to zero in different rates as vertical bar x vertical bar -> infinity in different directions. In un-mixed norm case, the result recovers available results. With some additional geometric assumptions on the supports of solutions in weighted mixed-norm Lebesgue spaces, this work also provides several other important Liouville type theorems of solutions in weighted mixed-norm Lebesgue spaces.

In this paper, we study the asymptotic behavior of solutions to the incompressible magnetohydrodynamic (MHD) equations in an exterior domain. We will show that, under some assumption, any nontrivial velocity field u and magnetic field h obey a minimal decaying rate exp(−C|x|2 log |x|) at infinity. Our proof is based on appropriate Carleman estimates. As a consequence, we establish a Liouville-type result for the three dimensional incompressible MHDs in an exterior domain.

We prove some existence results for the fractional Yamabe problem in the case that the boundary manifold is umbilic, thus covering some of the cases not considered by Gonzalez and Qing. These are inspired by the work of Coda-Marques on the boundary Yamabe problem but, in addition, a careful understanding of the behavior at infinity for asymptotically hyperbolic metrics is required.

In this paper, we consider the Cauchy problem of hydrodynamic flow of nematic liquid crystals.We get the uniqueness of the mild solution in C([0, T ); L n ).

Recently, Du, Guth and Li showed that the Schrödinger operator $e^{it\Delta }$ satisfies $\lim_{t\rightarrow 0}e^{it\Delta }f=f$ almost everywhere for all $f\in H^{s}(\mathbb{R}^{2})$, provided that $s>1/3$. In this paper, we discuss the rate of convergence on $e^{it\Delta }(f)$ by assuming more regularity on $f$. At $n=2$, our result can be viewed as an application of the Du–Guth–Li theorem. We also address the same issue on the cases $n=1$ and $n>2$.

本文考虑向列相液晶的动力学方程的Cauchy问题,并给出温和解在C([0,T);Ln)中解的唯一性.

In this paper, we consider the solutions of the relaxed Q-tensor flow in \({\mathbb{R}^3}\) with small parameter \({\epsilon}\). We show that the limiting map is the so-called harmonic map flow. As a consequence, we present a new proof for the global existence of a weak solution for the harmonic map flow in three dimensions as in [18, 23], where the Ginzburg–Landau approximation approach was used.

In this paper, we consider the Landau-Lifshitz equation of the ferromagnetic spin chain from $\R^2$ to the unit sphere $S^2$ under the general Oseen-Frank energy. We obtain global existence and uniqueness of weak solutions for large energy data; moreover, the number of singular points is finite.