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Propagation of singularities for the wave equation on manifolds with corners

András Vasy (2004/2005)

Séminaire Équations aux dérivées partielles

In this talk we describe the propagation of 𝒞 ∞ and Sobolev singularities for the wave equation on 𝒞 ∞ manifolds with corners M equipped with a Riemannian metric g . That is, for X = M × ℝ t , P = D t 2 - Δ M , and u ∈ H loc 1 ( X ) solving P u = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WF b ( u ) is a union of maximally extended generalized broken bicharacteristics. This result is a 𝒞 ∞ counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary,...

Propagation of singularities in many-body scattering in the presence of bound states

András Vasy (1999)

Journées équations aux dérivées partielles

In these lecture notes we describe the propagation of singularities of tempered distributional solutions u ∈ 𝒮 ' of ( H - λ ) u = 0 , where H is a many-body hamiltonian H = Δ + V , Δ ≥ 0 , V = ∑ a V a , and λ is not a threshold of H , under the assumption that the inter-particle (e.g. two-body) interactions V a are real-valued polyhomogeneous symbols of order - 1 (e.g. Coulomb-type with the singularity at the origin removed). Here the term “singularity” provides a microlocal description of the lack of decay at infinity. Our result is then that the...

Propagation of uniform Gevrey regularity of solutions to evolution equations

Todor Gramchev, Ya-Guang Wang (2003)

Banach Center Publications

We investigate the propagation of the uniform spatial Gevrey G σ , σ ≥ 1, regularity for t → +∞ of solutions to evolution equations like generalizations of the Euler equation and the semilinear Schrödinger equation with polynomial nonlinearities. The proofs are based on direct iterative arguments and nonlinear Gevrey estimates.

Propagation of weak discontinuities for quasilinear hyperbolic systems with coefficients functionally dependent on solutions

Małgorzata Zdanowicz, Zbigniew Peradzyński (2013)

Annales Polonici Mathematici

The propagation of weak discontinuities for quasilinear systems with coefficients functionally dependent on the solution is studied. We demonstrate that, similarly to the case of usual quasilinear systems, the transport equation for the intensity of weak discontinuity is quadratic in this intensity. However, the contribution from the (nonlocal) functional dependence appears to be in principle linear in the jump intensity (with some exceptions). For illustration, several examples, including two hyperbolic...

Propagation through trapped sets and semiclassical resolvent estimates

Kiril Datchev, András Vasy (2012)

Annales de l’institut Fourier

Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single isolated periodic trajectory. This is applied to obtain microlocal resolvent estimates with no loss compared to the nontrapping setting.

Proper orthogonal decomposition for optimality systems

Karl Kunisch, Stefan Volkwein (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

Proper orthogonal decomposition (POD) is a powerful technique for model reduction of non-linear systems. It is based on a Galerkin type discretization with basis elements created from the dynamical system itself. In the context of optimal control this approach may suffer from the fact that the basis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. A method is proposed which avoids this problem of unmodelled...

Properties of a quasi-uniformly monotone operator and its application to the electromagnetic $p$-$\text {curl}$ systems

Chang-Ho Song, Yong-Gon Ri, Cholmin Sin (2022)

Applications of Mathematics

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation $Au=b$. We prove that if $A$ is a quasi-uniformly monotone and hemi-continuous operator, then $A^{-1}$ is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence...

Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations

Ciprian Foias, Ricardo M. S. Rosa, Roger Temam (2013)

Annales de l’institut Fourier

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...

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