Local-in-time Strichartz estimates for the Maxwell system
A talk in the Oberseminar Analysis series by
Roland Schnaubelt from Karlsruhe
| Abstract: | Strichartz inequalities quantify the dispersive behavior of linear
wave-type equations: an increased space integrability can be obtained at
the prize of decreased time integrability. Such estimates play a crucial
role in the investigation of nonlinear variants of the equations. For
quasilinear problems one actually needs Strichartz estimates for
time-depending low-regularity coefficients. For scalar wave equations a quite complete theory is available, at least locally in time. Sharp results on the full space were obtained about 20 years ago by Tataru. Here a regularity loss occurs if the coefficients belong to $C^s$ with $s<2$. On the other hand, almost nothing was known on the Maxwell systems until recently. Here we have established sharp results for the isotropic case (with scalar coefficients), which have been applied to the local wellposedness theory of the corresponding quasilinear system. We use results and methods by Tataru. However, the kernel of curl operator in the problem poses significant new difficulties. Our approach extends to some partially non-isotropic cases, but in the fully non-isotropic case the situation changes drastically since the `light cone' becomes singular. So far we have achieved non-sharp estimates here. This is partly joint work with Robert Schippa (Karlsruhe) and partly by him alone. I will briefly mention work in progress by Robert Schippa and Nicholas Burq (Paris) on the domain case. Zoom Meeting ID: 986 0677 6055 Passcode: 430747 $\href{https://uni-bielefeld.zoom.us/j/98606776055?pwd=azBFdUVaQnJMU01tWjZUVEdxN2x5Zz09}{\textbf{Join Zoom Meeting}}$ Within the CRC this talk is associated to the project(s): A1 |