### Induced trajectories and approximate inertial manifolds

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In this article we apply the optimal and the robust control theory to the sine-Gordon equation. In our case the control is given by the boundary conditions and we work in a finite time horizon. We present at the beginning the optimal control problem and we derive a necessary condition of optimality and we continue by formulating a robust control problem for which existence and uniqueness of solutions are derived.

The objective of this work is to obtain theoretical estimates on the large and small scales for geophysical flows. Firstly, we consider the shallow water problem in the one-dimensional case, then in the two-dimensional case. Finally we consider geophysical flows under the hydrostatic hypothesis and the Boussinesq approximation. Scale separation is based on Fourier series, with N models in each spatial direction, and the choice of a cut-off level N < N to define large and small scales. We...

This article is concerned with the nonlinear singular perturbation problem due to small diffusivity in nonlinear evolution equations of Chaffee-Infante type. The boundary layer appearing at the boundary of the domain is fully described by a corrector which is “explicitly" constructed. This corrector allows us to obtain convergence in Sobolev spaces up to the boundary.

We analyze the equation of water vapor content in the atmosphere taking into account the saturation phenomenon. This equation is considered alone or coupled with the equation describing the evolution of the temperature $T$. The concentration of water vapor $q$ belongs to the interval $[0,1]$ and the saturation concentration ${q}_{s}\in (0,1)$ is the threshold after which the vapor condensates and becomes water (rain). The equation for $q$ (as well as the coupled $q-T$ system) thus accounts for possible change of phase.

A new set of nonlocal boundary conditions is proposed for the higher modes of the 3D inviscid primitive equations. Numerical schemes using the splitting-up method are proposed for these modes. Numerical simulations of the full nonlinear primitive equations are performed on a nested set of domains, and the results are discussed.

On a bounded smooth domain $\mathrm{\Omega}\subset {\mathbb{R}}^{3}$, we consider the generalized oscillon equation $${\partial}_{tt}u(x,t)+\omega (t){\partial}_{t}u(x,t)-\mu (t)\mathrm{\Delta}u(x,t)+{V}^{\prime}(u(x,t))=0,x\in \mathrm{\Omega}\subset {\mathbb{R}}^{3},t\in \mathbb{R}$$ with Dirichlet boundary conditions, where $\omega $ is a time-dependent damping, $\mu $ is a time-dependent squared speed of propagation, and $V$ is a nonlinear potential of critical growth. Under structural assumptions on $\omega $ and $\mu $ we establish the existence of a pullback global attractor $\mathcal{A}=\mathcal{A}(t)$ in the sense of [1]. Under additional assumptions on $\mu $, which include the relevant physical cases, we obtain optimal regularity of the pull-back global...

This work is devoted to the concept of statistical solution of the Navier-Stokes equations, proposed as a rigorous mathematical object to address the fundamental concept of ensemble average used in the study of the conventional theory of fully developed turbulence. Two types of statistical solutions have been proposed in the 1970’s, one by Foias and Prodi and the other one by Vishik and Fursikov. In this article, a new, intermediate type of statistical solution is introduced and studied. This solution...

New avenues are explored for the numerical study of the two dimensional inviscid hydrostatic primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically plausible boundary conditions for the system of equations. Flows above a mountain are classically treated by the so-called method of terrain following coordinate system. We avoid this discretization method which induces errors in the discretization of tangential derivatives near the topography....

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