Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

Michael Ruzhansky,Niyaz Tokmagambetov

doi:10.1007/s00205-017-1152-x

Abstract

Given a Hilbert space {mathcal{H}}, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on {mathcal{H}}. We consider the cases when the time-dependent propagation speed is regular, Hölder, and distributional. We also consider cases when it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of “very weak solutions” to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the harmonic and anharmonic oscillators, the Landau Hamiltonian on {mathbb{R}^n}, uniformly elliptic operators of different orders on domains, Hörmander’s sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others.

Highlights

Let L be a densely defined linear operator with the discrete spectrum {λξ ≥ 0 : ξ ∈ I}on a separable Hilbert space H
For a non-negative function a = a(t) ≥ 0 and for the source term f = f (t) ∈ H, we are interested in the well-posedness of the Cauchy problem for the operator L with the propagation speed given by a:
In this paper we consider the following situations: (II.1) The coefficient a ≥ a0 > 0 is a strictly positive distribution and the Cauchy data u0, u1 and the source term f (t) belong to the L-Sobolev spaces HLs for some s ∈ R. In this case we prove the existence and uniqueness of Sobolevtype very weak solutions, and their consistency with cases (I.1)–(I.4) when we know that stronger solutions exist

Summary

Introduction

(II.1) The coefficient a ≥ a0 > 0 is a strictly positive distribution and the Cauchy data u0, u1 and the source term f (t) belong to the L-Sobolev spaces HLs for some s ∈ R In this case we prove the existence and uniqueness of Sobolevtype very weak solutions, and their consistency with cases (I.1)–(I.4) when we know that stronger solutions exist. In our results below, concerning the Cauchy problem (1.1), we first carry out analysis in the strictly hyperbolic case a(t) ≥ a0 > 0, a ∈ C1([0, T ]) This is the regular strictly hyperbolic type case when we obtain the well-posedness in Sobolev spaces HLs associated to the operator L: for any s ∈ R, we set HLs := f ∈ HL−∞ : Ls/2 f ∈ H ,. We refer to [25] for the historical review of this topic

Examples

Landau Hamiltonian in 2D

Harmonic and Anharmonic Oscillators

Higher Dimensional Hamiltonian

Regular Elliptic Boundary Value Problems

Sums of Squares on Compact Lie Groups

Operators on Manifolds with Boundary

Differential Operators with Strong Regular Boundary Conditions

L–Fourier Analysis

Proofs of Part I

Existence of Very Weak Solutions

Consistency with the Classical Well–Posedness