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I gave recently some lectures (Berlin, Rutgers, College Park, Miami) about the construction of local canonical foliations of observers in Einstein spacetimes of general relativity when the curvature is solely assumed to be bounded and no assumption on its derivatives is made. In this joint work with B.-L. Chen, under geometric bounds on the curvature and injectivity radius near the observer, I proved that there exist a CMC (constant mean curvature) foliation as well as CMC–harmonic coordinates. These objects are defined in geodesic balls with definite size depending only on the assumed bounds, and the components of the Lorentzian metric has optimal regularity in these coordinates. The proof combines geometric estimates (Jacobi field, comparison theorems) and quantitative estimates for nonlinear elliptic equations with low regularity.

The lectures are based on the following two papers:

B.-L. Chen and P.G. LeFloch, Injectivity radius estimates for Lorentzian manifolds, Commun. Math. Phys. 278 (2008), 679–713.

B.-L. Chen and P.G. LeFloch, Local foliations and optimal regularity of Einstein spacetimes, J. Geom. Phys. 59 (2009), 913–-941.

I gave a series of lectures in Oslo and in Rio de Janeiro, which reviewed various issues about undercompressive shocks and their selection via a kinetic relation. The slides of these lectures can be downloaded at this link. They were also used for one-hour lectures given in Lyon, Nancy,  Minneapolis and Bilbao.

See also here the video of the lecture at Minneapolis in July 2009.

Abstract of the lectures. I discuss the existence and properties of small-scale dependent shock waves to nonlinear hyperbolic systems, with an emphasis on the theory of nonclassical entropy solutions involving undercompressive shocks. Regularization-sensitive structures often arise in continuum physics, especially in flows of complex fluids or solids. The so-called kinetic relation was introduced for van der Waals fluids and austenite-martensite boundaries (Abeyaratne, Knowles, Truskinovsky) and nonlinear hyperbolic systems (LeFloch) to characterize the correct dynamics of subsonic phase boundaries and undercompressive shocks, respectively. The role of a single entropy inequality is essential for these problems and is tied to the regularization associated with higher-order underlying models –which take into account additional physics and provide a description of small-scale effects. In the last fifteen years, analytical and numerical techniques were developed, beginning with the construction of nonclassical Riemann solvers, which were applied to tackle the initial-value problem via the Glimm scheme. Total variation functionals adapted to nonclassical entropy solutions were constructed. On the other hand, the role of traveling waves in selecting the proper shock dynamics was stressed: traveling wave solutions (to the Navier-Stokes-Korteweg system, for instance) determine the relevant kinetic relation –as well as the relevant family of paths in the context of nonconservative systems. Several physical applications were pursued: (hyperbolic-elliptic) equations of van der Waals fluids, model of thin liquid films, generalized Camassa-Holm equations, etc. Importantly, finite difference schemes with controled dissipation based on the equivalent equation were designed and the corresponding kinetic functions computed numerically. Consequently, `several shock wave theories’ are now available to encompass the variety of phenomena observed in complex flows.

References.

1993: P.G. LeFloch, Propagating phase boundaries. Formulation of the problem and existence via the Glimm scheme, Arch. Rational Mech. Anal. 123, 153–197.

1997: B.T. Hayes and P.G. LeFloch, Nonclassical shocks and kinetic relations. Scalar conservation laws, Arch. Rational Mech. Anal. 139, 1–56.

2002: P.G. LeFloch, Hyperbolic Systems of Conservation Laws. The theory of classical and nonclassical shock waves, Lectures in Mathematics, ETH Zurich, Birkhauser.

2004: N. Bedjaoui and P.G. LeFloch, Diffusive-dispersive traveling waves and kinetic relations. V. Singular diffusion and dispersion terms, Proc. Royal Soc. Edinburgh 134A, 815–844.

2008: P.G. LeFloch and M. Mohamadian, Why many shock wave theories are necessary. Fourth-order models, kinetic functions, and equivalent equations, J. Comput. Phys. 227, 4162–4189.

Philippe LeFloch, DIRECTOR OF RESEARCH AT CNRS Email address: pglefloch [at] gmail.com

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