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Discrete & Continuous Dynamical Systems - B

2016 , Volume 21 , Issue 9

Special issue dedicated to Björn Schmalfuß on the occasion of his 60th birthday

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Preface
Tomás Caraballo, María J. Garrido-Atienza and Wilfried Grecksch
2016, 21(9): i-ii doi: 10.3934/dcdsb.201609i +[Abstract](387) +[PDF](91.4KB)
Abstract:
It is a great honor and pleasure to dedicate this special issue of the journal Discrete and Continuous Dynamical Systems, Series B, to our colleague and friend Björn Schmalfuß, on the occasion of his 60th birthday.

For more information please click the “Full Text” above.
Statistical properties of stochastic 2D Navier-Stokes equations from linear models
Hakima Bessaih and Benedetta Ferrario
2016, 21(9): 2927-2947 doi: 10.3934/dcdsb.2016080 +[Abstract](501) +[PDF](471.9KB)
Abstract:
A new approach to the old-standing problem of the anomaly of the scaling exponents of nonlinear models of turbulence has been proposed and tested through numerical simulations. This is achieved by constructing, for any given nonlinear model, a linear model of passive advection of an auxiliary field whose anomalous scaling exponents are the same as the scaling exponents of the nonlinear problem.
    In this paper, we investigate this conjecture for the 2D Navier-Stokes equations driven by an additive noise. In order to check this conjecture, we analyze the coupled system Navier-Stokes/linear advection system in the unknowns $(u,w)$. We introduce a parameter $\lambda$ which gives a system $(u^\lambda,w^\lambda)$; this system is studied for any $\lambda$ proving its well posedness and the uniqueness of its invariant measure $\mu^\lambda$.
    The key point is that for any $\lambda \neq 0$ the fields $u^\lambda$ and $w^\lambda$ have the same scaling exponents, by assuming universality of the scaling exponents to the force. In order to prove the same for the original fields $u$ and $w$, we investigate the limit as $\lambda \to 0$, proving that $\mu^\lambda$ weakly converges to $\mu^0$, where $\mu^0$ is the only invariant measure for the joint system for $(u,w)$ when $\lambda=0$.
Equi-attraction and continuity of attractors for skew-product semiflows
Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa and José A. Langa
2016, 21(9): 2949-2967 doi: 10.3934/dcdsb.2016081 +[Abstract](662) +[PDF](400.6KB)
Abstract:
In this paper we prove the equivalence between equi-attraction and continuity of attractors for skew-product semi-flows, and equi-attraction and continuity of uniform and cocycle attractors associated to non-autonomous dynamical systems. To this aim proper notions of equi-attraction have to be introduced in phase spaces where the driving systems depend on a parameter. Results on the upper and lower-semicontinuity of uniform and cocycle attractors are relatively new in the literature, as a deep understanding of the internal structure of these sets is needed, which is generically difficult to obtain. The notion of lifted invariance for uniform attractors allows us to compare the three types of attractors and introduce a common framework in which to study equi-attraction and continuity of attractors. We also include some results on the rate of attraction to the associated attractors.
Synchronization in coupled stochastic sine-Gordon wave model
Igor Chueshov, Peter E. Kloeden and Meihua Yang
2016, 21(9): 2969-2990 doi: 10.3934/dcdsb.2016082 +[Abstract](486) +[PDF](550.1KB)
Abstract:
The asymptotic synchronization at the level of global random attractors is investigated for a class of coupled stochastic second order in time evolution equations. The main focus is on sine-Gordon type models perturbed by additive white noise. The model describes distributed Josephson junctions. The analysis makes extensive use of the method of quasi-stability.
The transport equation and zero quadratic variation processes
Jorge Clarke, Christian Olivera and Ciprian Tudor
2016, 21(9): 2991-3002 doi: 10.3934/dcdsb.2016083 +[Abstract](609) +[PDF](400.6KB)
Abstract:
We analyze the transport equation driven by a zero quadratic variation process. Using the stochastic calculus via regularization and the Malliavin calculus techniques, we prove the existence, uniqueness and absolute continuity of the law of the solution. As an example, we discuss the case when the noise is a Hermite process.
Weak synchronization for isotropic flows
Michael Cranston, Benjamin Gess and Michael Scheutzow
2016, 21(9): 3003-3014 doi: 10.3934/dcdsb.2016084 +[Abstract](546) +[PDF](405.7KB)
Abstract:
We study Brownian flows on manifolds for which the associated Markov process is strongly mixing with respect to an invariant probability measure and for which the distance process for each pair of trajectories is a diffusion $r$. We provide a sufficient condition on the boundary behavior of $r$ at $0$ which guarantees that the statistical equilibrium of the flow is almost surely a singleton and its support is a weak point attractor. The condition is fulfilled in the case of negative top Lyapunov exponent, but it is also fulfilled in some cases when the top Lyapunov exponent is zero. Particular examples are isotropic Brownian flows on $S^{d-1}$ as well as isotropic Ornstein-Uhlenbeck flows on $\mathbb{R}^d$.
An integral inequality for the invariant measure of some finite dimensional stochastic differential equation
Giuseppe Da Prato
2016, 21(9): 3015-3027 doi: 10.3934/dcdsb.2016085 +[Abstract](602) +[PDF](386.0KB)
Abstract:
We prove an integral inequality for the invariant measure $\nu$ of a stochastic differential equation with additive noise in a finite dimensional space $H=\mathbb R^d$. As a consequence, we show that there exists the Fomin derivative of $\nu$ in any direction $z\in H$ and that it is given by $v_z=\langle D\log\rho,z\rangle$, where $\rho$ is the density of $\nu$ with respect to the Lebesgue measure. Moreover, we prove that $v_z\in L^p(H,\nu)$ for any $p\in[1,\infty)$. Also we study some properties of the gradient operator in $L^p(H,\nu)$ and of his adjoint.
Mean field limit with proliferation
Franco Flandoli and Matti Leimbach
2016, 21(9): 3029-3052 doi: 10.3934/dcdsb.2016086 +[Abstract](590) +[PDF](445.1KB)
Abstract:
An interacting particle system with long range interaction is considered. Particles, in addition to the interaction, proliferate with a rate depending on the empirical measure. We prove convergence of the empirical measure to the solution of a parabolic equation with non-local nonlinear transport term and proliferation term of logistic type.
Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions
Hongjun Gao and Chengfeng Sun
2016, 21(9): 3053-3073 doi: 10.3934/dcdsb.2016087 +[Abstract](669) +[PDF](456.4KB)
Abstract:
Three dimensional primitive equations with a small multiplicative noise are studied in this paper. The existence and uniqueness of solutions with small initial value in a fixed probability space are obtained. The proof is based on Galerkin approximation, Itô's formula and weak convergence methods.
Semilinear stochastic equations with bilinear fractional noise
María J. Garrido-Atienza, Bohdan Maslowski and Jana  Šnupárková
2016, 21(9): 3075-3094 doi: 10.3934/dcdsb.2016088 +[Abstract](570) +[PDF](428.8KB)
Abstract:
In the paper, we study existence and uniqueness of solutions to semilinear stochastic evolution systems, driven by a fractional Brownian motion with bilinear noise term, and the long time behavior of solutions to such equations. For this purpose, we study at first the random evolution operator defined by the corresponding bilinear equation which is later used to define the mild solution of the semilinear equation. The mild solution is also shown to be weak in the PDE sense. Furthermore, the asymptotic behavior is investigated by using the Random Dynamical Systems theory. We show that the solution generates a random dynamical system that, under appropriate stability and compactness conditions, possesses a random attractor.
Linear approximation of nonlinear Schrödinger equations driven by cylindrical Wiener processes
Wilfried Grecksch and Hannelore Lisei
2016, 21(9): 3095-3114 doi: 10.3934/dcdsb.2016089 +[Abstract](464) +[PDF](467.0KB)
Abstract:
In this paper the existence and uniqueness of the solution for a stochastic nonlinear Schrödinger equation, which is perturbed by a cylindrical Wiener process is investigated. The existence of the variational solution and of the generalized weak solution are proved by using sequences of successive approximations, which are the solutions of certain linear problems.
Taylor schemes for rough differential equations and fractional diffusions
Yaozhong Hu, Yanghui Liu and David Nualart
2016, 21(9): 3115-3162 doi: 10.3934/dcdsb.2016090 +[Abstract](496) +[PDF](701.9KB)
Abstract:
In this paper, we study two variations of the time discrete Taylor schemes for rough differential equations and for stochastic differential equations driven by fractional Brownian motions. One is the incomplete Taylor scheme which excludes some terms of an Taylor scheme in its recursive computation so as to reduce the computation time. The other one is to add some deterministic terms to an incomplete Taylor scheme to improve the mean rate of convergence. Almost sure rate of convergence and $L_p$-rate of convergence are obtained for the incomplete Taylor schemes. Almost sure rate is expressed in terms of the Hölder exponents of the driving signals and the $L_p$-rate is expressed by the Hurst parameters. Both the almost sure and the $L_{p}$-convergence rates can be computed explicitly in terms of the parameters and the number of terms included in the incomplete scheme. In this way we can design the best incomplete schemes for the almost sure or the $L_p$-convergence. As in the smooth case, general Taylor schemes are always complicated to deal with. The incomplete Taylor scheme is even more sophisticated to analyze. A new feature of our approach is the explicit expression of the error functions which will be easier to study. Estimates for multiple integrals and formulas for the iterated vector fields are obtained to analyze the error functions and then to obtain the rates of convergence.
Approximation for random stable manifolds under multiplicative correlated noises
Tao Jiang, Xianming Liu and Jinqiao Duan
2016, 21(9): 3163-3174 doi: 10.3934/dcdsb.2016091 +[Abstract](552) +[PDF](418.8KB)
Abstract:
The usual Wong-Zakai approximation is about simulating individual solutions of stochastic differential equations(SDEs). From the perspective of dynamical systems, it is also interesting to approximate random invariant manifolds which are geometric objects useful for understanding how complex dynamics evolve under stochastic influences. We study a Wong-Zakai type of approximation for the random stable manifold of a stochastic evolutionary equation with multiplicative correlated noise. Based on the convergence of solutions on the invariant manifold, we approximate the random stable manifold by the invariant manifolds of a family of perturbed stochastic systems with smooth correlated noise (i.e., an integrated Ornstein-Uhlenbeck process). The convergence of this approximation is established.
Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process
Isabelle Kuhwald and Ilya Pavlyukevich
2016, 21(9): 3175-3190 doi: 10.3934/dcdsb.2016092 +[Abstract](604) +[PDF](948.1KB)
Abstract:
We study a bistable gradient system perturbed by a stable-like additive process with a periodically varying stability index. Among a continuum of intrinsic time scales determined by the values of the stability index we single out the characteristic time scale on which the system exhibits the metastable behaviour, namely it behaves like a time discrete two-state Markov chain.
Takens theorem for random dynamical systems
Weigu Li and Kening Lu
2016, 21(9): 3191-3207 doi: 10.3934/dcdsb.2016093 +[Abstract](726) +[PDF](405.9KB)
Abstract:
In this paper, we study random dynamical systems with partial hyperbolic fixed points and prove the smooth conjugacy theorems of Takens type based on their Lyapunov exponents.
Decomposition of stochastic flows generated by Stratonovich SDEs with jumps
Alison M. Melo, Leandro B. Morgado and Paulo R. Ruffino
2016, 21(9): 3209-3218 doi: 10.3934/dcdsb.2016094 +[Abstract](480) +[PDF](362.7KB)
Abstract:
Consider a manifold $M$ endowed locally with a pair of complementary distributions $\Delta^H \oplus \Delta^V=TM$ and let $\text{Diff}(\Delta^H, M)$ and $\text{Diff}(\Delta^V, M)$ be the corresponding Lie subgroups generated by vector fields in the corresponding distributions. We decompose a stochastic flow with jumps, up to a stopping time, as $\varphi_t = \xi_t \circ \psi_t$, where $\xi_t \in \text{Diff}(\Delta^H, M)$ and $\psi_t \in \text{Diff}(\Delta^V, M)$. Our main result provides Stratonovich stochastic differential equations with jumps for each of these two components in the corresponding infinite dimensional Lie groups. We present an extension of the Itô-Ventzel-Kunita formula for stochastic flows with jumps generated by classical Marcus equation (as in Kurtz, Pardoux and Protter [11]). The results here correspond to an extension of Catuogno, da Silva and Ruffino [4], where this decomposition was studied for the continuous case.
Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and non-explosion results
Michael Röckner, Jiyong Shin and Gerald Trutnau
2016, 21(9): 3219-3237 doi: 10.3934/dcdsb.2016095 +[Abstract](527) +[PDF](493.0KB)
Abstract:
Using elliptic regularity results in weighted spaces, stochastic calculus and the theory of non-symmetric Dirichlet forms, we first show weak existence of non-symmetric distorted Brownian motion for any starting point in some domain $E$ of $\mathbb{R}^d$, where $E$ is explicitly given as the points of strict positivity of the unique continuous version of the density to its invariant measure. This non-symmetric distorted Brownian motion is also proved to be strong Feller. Non-symmetric distorted Brownian motion is a singular diffusion, i.e. a diffusion that typically has an unbounded and discontinuous drift. Once having shown weak existence, we obtain from a result of [13] that the constructed weak solution is indeed strong and weakly as well as pathwise unique up to its explosion time. As a consequence of our approach, we can use the theory of Dirichlet forms to prove further properties of the solutions. For example, we obtain new non-explosion criteria for them. We finally present concrete existence and non-explosion results for non-symmetric distorted Brownian motion related to a class of Muckenhoupt weights and corresponding divergence free perturbations.
Global attractors for $p$-Laplacian differential inclusions in unbounded domains
Jacson Simsen and José Valero
2016, 21(9): 3239-3267 doi: 10.3934/dcdsb.2016096 +[Abstract](726) +[PDF](596.7KB)
Abstract:
In this work we consider a differential inclusion governed by a p-Laplacian operator with a diffusion coefficient depending on a parameter in which the space variable belongs to an unbounded domain. We prove the existence of a global attractor and show that the family of attractors behaves upper semicontinuously with respect to the diffusion parameter. Both autonomous and nonautonomous cases are studied.
Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises
Jiahui Zhu and Zdzisław Brzeźniak
2016, 21(9): 3269-3299 doi: 10.3934/dcdsb.2016097 +[Abstract](708) +[PDF](556.8KB)
Abstract:
We study a class of abstract nonlinear stochastic equations of hyperbolic type driven by jump noises, which covers both beam equations with nonlocal, nonlinear terms and nonlinear wave equations. We derive an Itô formula for the local mild solution which plays an important role in the proof of our main results. Under appropriate conditions, we prove the non-explosion and the asymptotic stability of the mild solution.

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