ISSN:
1078-0947
eISSN:
1553-5231
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Discrete & Continuous Dynamical Systems
October 1997 , Volume 3 , Issue 4
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I consider discretized random perturbations of hyperbolic dynamical systems and prove that when perturbation parameter tends to zero invariant measures of corresponding Markov chains converge to the Sinai-Bowen-Ruelle measure of the dynamical system. This provides a robust method for computations of such measures and for visualizations of some hyperbolic attractors by modeling randomly perturbed dynamical systems on a computer. Similar results are true for discretized random perturbations of maps of the interval satisfying the Misiurewicz condition considered in [KK].
This paper is concerned with the Cauchy problem
$(*) \quad \quad u_t+[F(u)]_x=g(t,x,u),\quad u(0,x)=\overline{u}(x),$
for a nonlinear $2\times 2$ hyperbolic system of
inhomogeneous balance laws
in one space dimension. As usual,
we assume that the system is strictly hyperbolic and that each
characteristic field is either
linearly degenerate or genuinely nonlinear.
Under suitable assumptions on $g$, we prove that there exists
$T>0$ such that, for every $\overline{u}$ with sufficiently small
total variation, the Cauchy problem ($*$)
has a unique "viscosity solution",
defined for $t\in [0,T]$,
depending continuously on the initial data.
We construct several approximate inertial manifolds for a weakly damped nonlinear Schrödinger equation. For that purpose, we introduce suitable smooth approximations for the solution of the equation and for its time derivatives.
In this article, we obtain the existence of inertial manifolds under time discretization based on their invariant property. In [1], the authors gave their existence by finding the fixed point of some inertial mapping defined by a sum of infinite series:
$ T_h^0\Phi(p)=\sum_{k=1}^{\infty}R(h)^kQF(p^{-k}+\Phi(p^{-k})) $
where $p^{-k}=(S^h_\Phi)^{-k}(p)$, see [1] for detailed definition. Here we get the existence by solving the following equation about $\Phi$:
$\Phi(S_\Phi^h(p))=R(h)[\Phi(p)+hQF(p+\Phi(p))] \mbox{ for }\forall p\in PH.$
See section 1 for further explanation which describes just the invariant property of inertial manifolds. Finally we prove the $C^1$ smoothness of inertial manifolds.
A global existence theorem for two semilinear diffusion equations is proved. The equations are coupled and the diffusion coefficients are not uniformly elliptic. They arise in the study of a simple zonally averaged climate model (See also [8, 9, 13, 14]). The sectoriality of the diffusion operator is proved with the help of a technique of F. Ali Mehmeti and S. Nicaise [2]. Some imbedding results for weighted Sobolev spaces and sign conditions for the nonlinearities allow the application of a result due to Amann [3], which proves the global result.
The nonexistence of positive solutions is discussed for $-\Delta_p u = a(x) u^{q-1}$ in $\Omega$, $u|_{\partial\Omega}= 0$, for the case where $a(x)$ is a bounded positive function and $\Omega$ is a strip-like domain such as $\Omega = \Omega_d \times \mathcal{R}^{N-d}$ with $\Omega_d$ bounded in $\mathcal{R}^{d}$. The existence of nontrivial solution of (E) is proved by Schindler for $q \in (p, p*)$ where $p*$ is Sobolev's critical exponent. Our method of proofs for nonexistence rely on the "Pohozaev-type inequality" (for $q \ge p*$); and on a new argument concerning the simplicity of the first eigenvalue for (generalized) eigenvalue problems combined with translation invariance of the domain (for $q \le p$).
This paper contains some existence and multiplicity results for periodic solutions of second order nonautonomous and nonsmooth Hamiltonian systems involving nonconvex superpotentials. This study is achieved by proving the existence of homoclinic solutions. These solutions are constructed as critical points of the corresponding nonconvex and nonsmooth energy functional.
We establish coincidence of major types of dimensions for a broad class of separable metric spaces with finite borel measures. To do this we introduce a new type of separable metric spaces, so called tight spaces, for which these dimensions coincide naturally. This class includes, for example, all manifolds of the curvature bounded from below and any their subsets with induced metric. In particular, we prove that Hentshel-Procaccia and Renyi spectra for dimensions are equal in tight spaces for any measure. We also give the examples that demonstrate that all known dimensions can differ for bad enough metric spaces.
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