Strictly hyperbolic Cauchy problems with coefficients low-regular in time and space

TL;DR

This paper establishes energy estimates for strictly hyperbolic equations with coefficients that are low-regular in time and space, extending well-posedness results under minimal regularity assumptions.

Contribution

It provides global $L^2$ energy estimates without loss of derivatives for hyperbolic equations with low-regularity coefficients in both time and space.

Findings

Energy estimates hold for coefficients in Zygmund class with low time regularity.

No loss of derivatives in $L^2$ energy estimates under specified regularity conditions.

Results extend well-posedness theory to less regular coefficients.

Abstract

We consider the strictly hyperbolic Cauchy problem \begin{align*} &D_t^m u - \sum\limits_{j = 0}^{m-1} \sum\limits_{|\gamma|+j = m} a_{m-j,\,\gamma}(t,\,x) D_x^\gamma D_t^j u = 0, \newline &D_t^{k-1}u(0,\,x) = g_k(x),\,k = 1,\,\ldots,\,m, \end{align*} for $(t,\,x) \in [0,\,T]\times \mathbb{R}^n$ with coefficients belonging to the Zygmund class $C^s_\ast$ in $x$ and having a modulus of continuity below Lipschitz in $t$. Imposing additional conditions to control oscillations, we obtain a global (on $[0,\,T]$) $L^2$ energy estimate without loss of derivatives for $s \geq \{1+\varepsilon,\,\frac{2m_0}{2-m_0}\}$, where $m_0$ is linked to the modulus of continuity of the coefficients in time.