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Register a new userRegularity and energy transfer for a nonlinear beam equation
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Added by
Giuseppe Maria Coclite
Publication date
2020
fields
Physics
and research's language is
English
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In this paper we study some key effects of a discontinuous forcing term in a fourth order wave equation on a bounded domain, modeling the adhesion of an elastic beam with a substrate through an elastic-breakable interaction. By using a spectral decomposition method we show that the main effects induced by the nonlinearity at the transition from attached to detached states can be traced in a loss of regularity of the solution and in a migration of the total energy through the scales.
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In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schrodinger equations (NLS) that we gave in cite{CM15APDE}. We consider a NLS with a Schrodinger operator with several eigenvalues, with corresponding families of small standing waves, and we show that any small energy solution converges to the orbit of a time periodic solution plus a scattering term. The novel idea is to consider the refined profile, a quasi--periodic function in time which almost solves the NLS and encodes the discrete modes of a solution. The refined profile, obtained by elementary means, gives us directly an optimal coordinate system, avoiding the normal form arguments in cite{CM15APDE}, giving us also a better understanding of the Fermi Golden Rule.
We establish Strichartz estimates for the radial energy-critical wave equation in 5 dimensions in similarity coordinates. Using these, we prove the nonlinear asymptotic stability of the ODE blowup in the energy space.
We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.
This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the meaningful objects are the fluxes, evaluated across domain boundaries over time intervals. The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting. It implies that a weak solution indeed satisfies the balance law. In fact, it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary.
We consider the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to $infty$. Both results rely on the Strichartz-type estimates for the corresponding linear equation.
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