Discrete and Continuous Dynamical Systems
August 2020 , Volume 40 , Issue 8
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In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of nonhomogeneous incompressible Navier-Stokes equations. The convergence is shown under various Sobolev norms, including the physically important space-time uniform norm, as well as the
This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non-autonomous SPDE by the finite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square
In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures and that one-sided one-dimensional DLR-Gibbs measures associated to a continuous translation invariant specifications are eigenmeasures of the transpose of the Ruelle operator. From the last claim one gets that for a continuous potential the concept of eigenprobability for the transpose of the Ruelle operator is equivalent to the concept of DLR probability.
Bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions, with respect to the eigenmeasures, are studied and the problem of existence of maximal positive eigenfunctions for them is considered. One of our main results in this direction is the existence of such positive eigenfunctions for Bowen's potential in the setting of a compact and metric alphabet. We also present a version of Dobrushin's Theorem in the setting of Thermodynamic Formalism.
In this work we deal with dynamically coherent partially hyperbolic diffeomorphisms whose central direction is two dimensional. We prove that in general the accessibility classes are topologically immersed manifolds. If, furthermore, the diffeomorphism satisfies certain bunching condition, then the accessibility classes are immersed
We study the fluctuations of ergodic sums using global and local specifications on periodic points. We obtain Lindeberg-type central limit theorems in both situations. As an application, when the system possesses a unique measure of maximal entropy, we show weak convergence of ergodic sums to a mixture of normal distributions. Our results suggest decomposing the variances of ergodic sums according to global and local sources.
This paper dealt with the existence of periodic waves for a perturbed quintic BBM equation by using geometric singular perturbation theory. By analyzing the perturbations of the Hamiltonian vector field with a hyperelliptic Hamiltonian of degree six, we proved that periodic wave solutions persist for sufficiently small perturbation parameter. It is also proved that the wave speed
This is the second installment in a series of papers aimed at generalizing symplectic capacities and homologies. We study symmetric versions of symplectic capacities for real symplectic manifolds, and obtain corresponding results for them to those of the first [
We study mean dimension of shifts of finite type defined on compact metric spaces and give its lower bound when the shift possesses a certain "periodic block" of arbitrarily large length. The result is applied to shift maps on generalized inverse limits with upper semi-continuous closed set-valued functions. In particular we obtain a refinement of some results due to Banič [
For non-critical extended Harper's model with Diophantine frequency, we establish the exponential decay of the upper bounds on the spectral gaps and prove the spectrum is homogeneous. Especially we give a relationship between the decaying rate and Lyapunov exponent in non-self-dual region.
We study the time-asymptotic behavior of solutions of the Schrödinger equation with nonlinear dissipation
In this paper, we consider the weighted problem
We give two type-homoclinic functions
Given a finite collection
1. For each
3. The expected topological pressure of the parameter
is independent of the choice of a
4. The function
is monotone decreasing and Lipschitz continuous.
We are interested in the Neumann problem of a 1D stationary Allen-Cahn equation with a nonlocal term. In our previous papers [
We propose a structure-preserving finite difference scheme for the Allen–Cahn equation with a dynamic boundary condition using the discrete variational derivative method [
We consider the convolution inequality
for given functions
By assuming certain local energy estimates on
This paper is concerned with the Cauchy problem of the Schrödinger-Hartree equation. Applying the profile decomposition of bounded sequence in
In this paper, we extend the nontangential maximal function estimates in
In this paper, we consider a semi-linear elliptic equation in
In this paper we deal with theoretical and numerical aspects of some nonlinear problems related to sandpile models. We introduce a purely discrete model for infinitely many particles interacting according to a toppling process on a uniform two-dimensional grid and prove the convergence of the solutions to a differential initial value problem.
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