Acta Mathematica Scientia最新文献

The nonlinear stability of plane parallel shear flows with respect to tilted perturbations is studied by energy methods. Tilted perturbation refers to the fact that perturbations form an angle (theta in (0,{pi over 2})) with the direction of the basic flows. By defining an energy functional, it is proven that plane parallel shear flows are unconditionally nonlinearly exponentially stable for tilted streamwise perturbation when the Reynolds number is below a certain critical value and the boundary conditions are either rigid or stress-free. In the case of stress-free boundaries, by taking advantage of the poloidal-toroidal decomposition of a solenoidal field to define energy functionals, it can be even shown that plane parallel shear flows are unconditionally nonlinearly exponentially stable for all Reynolds numbers, where the tilted perturbation can be either spanwise or streamwise.

In this paper, we consider a class of third-order nonlinear delay dynamic equations. First, we establish a Kiguradze-type lemma and some useful estimates. Second, we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero. Third, we obtain new oscillation criteria by employing the Pötzsche chain rule. Then, using the generalized Riccati transformation technique and averaging method, we establish the Philos-type oscillation criteria. Surprisingly, the integral value of the Philos-type oscillation criteria, which guarantees that all unbounded solutions oscillate, is greater than θ₄(t₁, T). The results of Theorem 3.5 and Remark 3.6 are novel. Finally, we offer four examples to illustrate our results.

We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping. The main novelty of this paper is two-fold. First, we prove the optimal decay rates of the second and third order spatial derivatives of the solution, which are the same as those of the heat equation, and in particular, are faster than ones of previous related works. Second, for well-chosen initial data, we also show that the lower optimal L² convergence rate of the k (∈ [0, 3])-order spatial derivatives of the solution is ({(1 + t)^{ - {{3 + 2k} over 4}}}). Therefore, our decay rates are optimal in this sense. The proofs are based on the Fourier splitting method, low-frequency and high-frequency decomposition, and delicate energy estimates.

Let X and Y be two normed spaces. Let ({cal U}) be a non-principal ultrafilter on ℕ. Let g: X → Y be a standard ε-phase isometry for some ε ≥ 0, i.e., g(0) = 0, and for all u, v ϵ X,

The mapping g is said to be a phase isometry provided that ε = 0. In this paper, we show the following universal inequality of g: for each ({u^ * } in {w^ * } - exp ,,||{u^ * }||{B_{{X^ * }}}), there exist a phase function ({sigma _{{u^ * }}}:X to { - 1,1} ) and φ ϵ Y* with (||varphi || = ||{u^ * }|| equiv alpha ) satisfying that

In particular, let X be a smooth Banach space. Then we show the following: (1) the universal inequality holds for all u* ∈ X*; (2) the constant ({5 over 2}) can be reduced to ({3 over 2}) provided that Y* is strictly convex; (3) the existence of such a g implies the existence of a phase isometry Θ: X → Y such that (Theta (u) = mathop {lim }limits_{n,{cal U}} {{g(nu)} over n}) provided that Y** has the w*-Kadec-Klee property (for example, Y is both reflexive and locally uniformly convex).

In this paper, we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms. We establish the global existence of mild solutions to this system with small initial data. In addition, we also obtain the Gevrey class regularity and the temporal decay rate of the solution.

In this paper, we consider the semilinear elliptic equation systems

where (Ngeqslant 3,,,alpha ,,,beta > 1,,alpha + beta < {2^ * },,{2^ * } = {{2N} over {N - 2}}) and Q_n are bounded given functions whose self-focusing cores {x ∈ ℍ^NQ_n(x) > 0} shrink to a set with finitely many points as n → ∞. Motivated by the work of Fang and Wang [13], we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points, and we build the localized concentrated bound state solutions for the above equation systems.

Motivated by some recent works on the topic of the Brown-Resnick process, we study the functional limit theorem for normalized pointwise maxima of dependent chi-processes. It is proven that the properly normalized pointwise maxima of those processes are attracted by the Brown-Resnick process.

This study introduces a pre-orthogonal adaptive Fourier decomposition (POAFD) to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre (generalized Poisson equation). As a first step, the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence, and, as a second step, makes use of the semigroup or the reproducing kernel property of each of the expanding entries. Experiments show the effectiveness and efficiency of the proposed series solutions.

Let B^H be a fractional Brownian motion with Hurst index ({1 over 2} le H < 1). In this paper, we consider the equation (called the Ornstein-Uhlenbeck process with a linear self-repelling drift)

where θ < 0, σ, v ∈ ℝ. The process is an analogue of self-attracting diffusion (Cranston, Le Jan. Math Ann, 1995, 303: 87–93). Our main aim is to study the large time behaviors of the process. We show that the solution X^H diverges to infinity as t tends to infinity, and obtain the speed at which the process X^H diverges to infinity as t tends to infinity.