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| Maximal regularity can often be used to obtain a priori estimates which give global existence results. For example, using maximal regularity it is possible to solve quasi-linear and fully nonlinear PDEs by elegant linearization techniques combined with the contraction mapping principle. In this talk we will prove a new mixed $L^{p}(0,T;L^{q}(\mathbb{R}^{d}))$ maximal regularity estimate for the PDE:
\[\begin{aligned}\left\{
\begin{array}{ll}
u'(t,x) +A(t,x)u(t,x) = f(t,x), & x\in \mathbb{R}^{d}, \ t\in [0,T] \\
u(0,x) = u_0(x), & x\in \mathbb{R}^{d}.
\end{array}
\right.
\end{aligned}\]
Here $A(t,x)$ is a second order elliptic differential operator.
The main novelty in our result is that the coefficients are merely measurable in time and we allow the full range $p,q\in(1,\infty)$ |
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