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arXiv · 1807.05811

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

Daniel Lorenz
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Abstract

We consider the strictly hyperbolic Cauchy problem \begin{align*} &D_t^m u - \sum\limits_{j = 0}^{m-1} \sum\limits_{|γ|+j = m} a_{m-j,\,γ}(t,\,x) D_x^γD_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.

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Daniel Lorenz. 2018-07-16. Strictly hyperbolic Cauchy problems with coefficients low-regular in time and space. https://arxiv.org/abs/1807.05811

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