Campuses:

Kinetic relations for undercompressive shocks. Physical, mathematical, and numerical issues

Thursday, July 23, 2009 - 11:00am - 11:50am
EE/CS 3-180
Philippe LeFloch (Université de Paris VI (Pierre et Marie Curie))
I will 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.
MSC Code: 
74A25
Keywords: