In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the $L^2$ critical and super-critical...

In this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction...

We consider the multifractal analysis for Birkhoff averages of continuous potentials on a class of non-conformal repellers corresponding to the self-affine limit sets studied by Lalley and Gatzouras. A conditional variational principle is given for the Hausdorff dimension of the set of points for which the Birkhoff averages converge to a given value. This extends a result of Barral and Mensi to certain non-conformal maps with a measure dependent Lyapunov exponent.

This article discusses a prey-predator system with cross-diffusion. We obtain multiple positive steady-state solutions of this system. More precisely, we prove that the set of positive steady-states possibly contains an S or ⊃-shaped branch with respect to a bifurcation parameter in the large cross-diffusion case. Next we give some criteria on the stability of these positive steady-states. Furthermore, we find the Hopf bifurcation point on the steady-state solution branch in a certain case. Our...

We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force $g(x,t)$. We assume that $g(x,t)$ is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if $g(x,t)$ is a quasiperiodic function with respect to $t$, then the attractor is a continuous image of a torus. Moreover the...

We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force g(x,t). We assume that g(x,t) is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if g(x,t) is a quasiperiodic function with respect to t, then the attractor is a continuous image...

We prove two new results about the Cauchy problem in the energy space for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadratic nonlinearities as well as a necessary and sufficient condition of global well-posedness for small energy...

Using Burgers’ equation with mixed Neumann–Dirichlet boundary conditions, we highlight a problem that can arise in the numerical approximation of nonlinear dynamical systems on computers with a finite precision floating point number system. We describe the dynamical system generated by Burgers’ equation with mixed boundary conditions, summarize some of its properties and analyze the equilibrium states for finite dimensional dynamical systems that are generated by numerical approximations of this...

We study a family of semilinear reaction-diffusion equations on spatial domains ${\Omega }_{\epsilon }$, ε > 0, in ${ℝ}^l$ lying close to a k-dimensional submanifold ℳ of ${ℝ}^l$. As ε → 0⁺, the domains collapse onto (a subset of) ℳ. As proved in [15], the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain limit phase space denoted by $H{¹}_s\left(\Omega \right)$. The definition of $H{¹}_s\left(\Omega \right)$, given in the above paper, is very abstract. One of the objectives of this paper is to give more manageable characterizations...

Let $S_i:{ℝ}^d\to {ℝ}^d$ for i = 1,..., N be contracting similarities, let $(p₁,...,p_N,p)$ be a probability vector and let ν be a probability measure on ${ℝ}^d$ with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on ${ℝ}^d$ such that $\mu ={\sum }_{i=1}^N{p}_i\mu \circ S_i^{-1}+p\nu$. We give satisfactory estimates for the lower and upper bounds of the $L^q$ spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular, we...

In this paper we study a general concept of nonuniform exponential dichotomy in mean square for stochastic skew-evolution semiflows in Hilbert spaces. We obtain a variant for the stochastic case of some well-known results, of the deterministic case, due to R. Datko: Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal., 3(1972), 428–445. Our approach is based on the extension of some techniques used in the deterministic case for the study of asymptotic behavior...

In this work we are interested in the study of controllability and stabilization of the linearized Benjamin-Ono equation with periodic boundary conditions, which is a generic model for the study of weakly nonlinear waves with nonlocal dispersion. It is well known that the Benjamin-Ono equation has infinite number of conserved quantities, thus we consider only controls acting in the equation such that the volume of the solution is conserved. We study also the stabilization with a feedback law which...