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Carnegie Mellon NSF logoCenter for Nonlinear Analysis
Mihaela Ignatova
Afilliation: Department of Mathematics, UC Riverside

Title: Quantitative unique continuation and complexity of solutions to parabolic equations

Abstract: In the first part of the talk, we address the strong unique continuation properties for 1D higher order parabolic partial differential equations with coefficients in the Gevrey class $G^{\sigma}$ for $\sigma>1$. We establish a quantitative estimate of unique continuation (observability estimate) under a mild assumption on the Gevrey exponent $\sigma$. Also, we improve the existing upper bounds on the size of the level sets of solutions and remove the restrictive analyticity requirement on the coefficients. Next, we consider the strong unique continuation problem for elliptic and parabolic equations in higher space dimensions. As an application, we provide a polynomial upper bound on the Hausdorff measure of the nodal (zero) sets of solutions in terms of the coefficients. In particular, we cover the case of the Navier-Stokes equations with non-analytic forcing. For this purpose, we provide Carleman-type inequalities with the same singular weight for the Laplacian and the heat operator. This is a joint work with Igor Kukavica.