# ON DEMAND International PDE Conference 2022 - DAY FOUR

**Prof. Denis Serre - Ecole Normale Supérieure de Lyon**

**Compensated Integrability and Conservation Laws**

Compensated Integrability, a tool from Functional Analysis, applies to positive semi-definite tensors whose row-wise Divergence is a finite measure. Div-free tensors occur naturally in various models of Mathematical Physics, as a consequence of N{\oe}ther's Theorem.

Somehow, Compensated Integrability is dual to Brenier's existence result for the "second BVP'' for the Monge--Ampère equation. It extends in a non-trivial manner the Gagliardo--Nirenberg--Sobolev Inequality, or the Isoperimetric Inequality. In the periodic situation, it expresses the Div-quasiconcavity of $A\mapsto(\det A)^{\frac1{n-1}}$ a non-concave function over ${\bf Sym}_n^+$), leading to a weak upper-semicontinuity result.

When it applies, C.I. yields dispersive (Strichartz-like) estimates. In Gas Dynamics, the internal energy cannot concentrate on zero-measure subsets. Other applications concern kinetic equations (Boltzman), mean-field models (Vlasov), molecular dynamics. The corresponding tensor is positive semi-definite whenever the particles interact pairwise according to a radial, repulsive force. In hard spheres dynamics, the Div-free tensor is supported by a graph, and a special form of C.I. is required.

Another relevant topic is that of multi-dimensional conservation laws, where C.I. allows us to extend Kru\v{z}kov's theory to $L^p$-data when $p$ is finite, under a non-degeneracy assumption (collaboration with L. Silvestre).

**Prof. Lawrence C. Evans - University of California, Berkeley**

**Compactness and the curvature of 3-webs**

**No presentation video published.*