# 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.

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: