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Register a new userMixing via controllability for randomly forced nonlinear dissipative PDEs
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We continue our study of the problem of mixing for a class of PDEs with very degenerate noise. As we established earlier, the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric holds under the hypothesis that the unperturbed equation has exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem. The proof uses a new idea that reduces the verification of a stability property to the investigation of a conditional random walk.
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We study ground state solutions for linear and nonlinear elliptic PDEs in $mathbb{R}^n$ with (pseudo-)differential operators of arbitrary order. We prove a general symmetry result in the nonlinear case as well as a uniqueness result for ground states in the linear case. In particular, we can deal with problems (e.,g. higher order PDEs) that cannot be tackled by usual methods such as maximum principles, moving planes, or Polya--Szego inequalities. Instead, we use arguments based on the Fourier transform and we apply a rigidity result for the Hardy-Littlewood majorant problem in $mathbb{R}^n$ recently obtained by the last two authors of the present paper.
We prove that well posed quasilinear equations of parabolic type, perturbed by bounded nondegenerate random forces, are exponentially mixing for a large class of random forces.
In this paper we review the basic results concerning the Wigner transform and then we completely solve the quantum forced harmonic/inverted oscillator in such a framework; eventually, the tunnel effect for the forced inverted oscillator is discussed.
An effective equation describes a weakly nonlinear wave field evolution governed by nonlinear dispersive PDEs emph{via} the set of its resonances in an arbitrary big but finite domain in the Fourier space. We consider the Schr{o}dinger equation with quadratic nonlinearity including small external random forcing/dissipation. An effective equation is deduced explicitly for each case of monomial quadratic nonlinearities $ u^2, , bar{u}u, , bar{u}^2$ and the sets of resonance clusters are studied. In particular, we demonstrate that the nonlinearity $bar{u}^2$ generates no 3-wave resonances and its effective equation is degenerate while in two other cases the sets of resonances are not empty. Possible implications for wave turbulence theory are briefly discussed.
Let $V(t) = e^{tG_b},: t geq 0,$ be the semigroup generated by Maxwells equations in an exterior domain $Omega subset {mathbb R}^3$ with dissipative boundary condition $E_{tan}- gamma(x) ( u wedge B_{tan}) = 0, gamma(x) > 0, forall x in Gamma = partial Omega.$ We study the case when $Omega = {x in {mathbb R^3}:: |x| > 1}$ and $gamma eq 1$ is a constant. We establish a Weyl formula for the counting function of the negative real eigenvalues of $G_b.$
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