Well-posedness results for hyperbolic operators with coefficients rapidly oscillating in time

TL;DR

This paper establishes well-posedness for second order hyperbolic operators with coefficients that oscillate rapidly in time, demonstrating results with no or finite derivative loss depending on spatial regularity.

Contribution

It provides new well-posedness results for hyperbolic equations with time-oscillating coefficients under minimal spatial regularity assumptions.

Findings

Well-posedness with no derivative loss for Lipschitz coefficients.

Finite derivative loss with linearly increasing loss for log-Lipschitz coefficients.

Handles coefficients with rapid oscillations in time near t=0.

Abstract

In the present paper, we consider second order strictly hyperbolic linear operators of the form $Lu\,=\,\partial_t^2u\,-\,{\rm div}\big(A(t,x)\nabla u\big)$, for $(t,x)\in[0,T]\times\mathbb{R}^n$. We assume the coefficients of the matrix $A(t,x)$ to be smooth in time on $\,]0,T]\times\mathbb{R}^n$, but rapidly oscillating when $t\to 0^+$; they match instead minimal regularity assumptions (either Lipschitz or log-Lipschitz regularity conditions) with respect to the space variable. Correspondingly, we prove well-posedness results for the Cauchy problem related to $L$, either with no loss of derivatives (in the Lipschitz case) or with a finite loss of derivatives, which is linearly increasing in time (in the log-Lipschitz case).