After a brief discussion of well‐known classical fields we formulate two principles: When the field equations are hyperbolic, particles move along rays like disturbances of the field; the waves associated with stable particles are exceptional. This means that these waves will not transform into shock waves. Both principles are applied to nonlinear electrodynamics. The starting point of the theory is a Lagrangian which is an arbitrary nonlinear function of the two electromagnetic invariants. We obtain the laws of propagation of photons and of charged particles, along with an anisotropic propagation of the wavefronts. The general ``exceptional'' Lagrangian is found. It reduces to the Lagrangian of Born and Infeld when some constant (probably simply connected with the Planck constant) vanishes. A nonsymmetric tensor is introduced in analogy to the Born‐Infeld theory, and finally, electromagnetic waves are compared with those of Einstein‐Schrodinger theory.

Nonlinear Electrodynamics: Lagrangians and Equations of Motion

We present some obvious physical requirements on gravitational avatars of nonlinear electrodynamics and illustrate them with explicit determinantal Born - Infeld - Einstein models. A related procedure, using compensating Weyl scalars, permits us to formulate conformally invariant versions of these systems as well.

/pdf/born-infeld-einstein-actions-3q8w4gdiyn.pdf

Born - Infeld - Einstein actions?

This paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables. We propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state. In addition, we show that a system of balance laws satisfies a Kawashima condition if and only if its first-order approximation, namely the hyperbolic-parabolic system derived through the Chapman-Enskog expansion, satisfies the corresponding Kawashima condition. The result is then applied to Bouchut’s discrete velocity BGK models approximating hyperbolic systems of conservation laws.

Entropy and Global Existence for Hyperbolic Balance Laws

The effect of damping on the large-time behavior of solutions to the Cauchy problem for the three-dimensional compressible Euler equations is studied. It is proved that damping prevents the develop...

https://www.math.ntnu.no/conservation/2003/005.pdf

Long Time Behavior of Solutions to the 3D Compressible Euler Equations with Damping

We consider the Cauchy problem for a general one-dimensional n×n hyperbolic symmetrizable system of balance laws. It is well known that, in many physical examples, for instance for the isentropic Euler system with damping, the dissipation due to the source term may prevent the shock formation, at least for smooth and small initial data. Our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions, and possibly to investigate their asymptotic behavior. For systems which are entropy dissipative, a quite natural generalization of the Kawashima condition for hyperbolic-parabolic systems can be given. In this paper, we first propose a general framework for this kind of problem, by using the so-called entropy variables. Then we go on to prove some general statements about the global existence of smooth solutions, under different sets of conditions. In particular, the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension. Our main tools will be some refined energy estimates and the use of a suitable version of the Kawashima condition.

Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy

We consider a system of N balance laws compatible with an entropy principle and convex entropy density. Using the special symmetric form induced by the main field, we define the concept of principal subsystem associated with the system. We prove that the 2
 N
 −2 principal subsystems are also symmetric hyperbolic and satisfy a subentropy law. Moreover we can verify that for each principal subsystem the maximum (minimum) characteristic velocity is not larger (smaller) than the maximum (minimum) characteristic velocity of the full system. These are the subcharacteristic conditions. We present some simple examples in the case of the Euler fluid. Then in the case of dissipative hyperbolic systems we consider an equilibrium principal subsystem and we discuss the consequences in the setting of extended thermodynamics. Finally in the moments approach to the Boltzmann equation we prove, as a consequence of the previous result, that the maximum characteristic velocity evaluated at the equilibrium state does not decrease when the number of moments increases.

Hyperbolic Principal Subsystems: Entropy Convexity and Subcharacteristic Conditions

In this paper we give a brief survey of the problem of shock structure solutions in fluid dynamics. For a generic system of balance laws compatible with an entropy principle and a convex entropy we prove that 
$C^1$
 solutions cannot exist when the shock velocity exceeds the maximum characteristic velocity in the equilibrium state in front of the shock. This is in agreement with a conjecture of Extended Thermodynamics.

On the shock structure problem for hyperbolic system of balance laws and convex entropy

On the evolution law of weak discontinuities for hyperbolic quasi-linear systems

A field is assumed to satisfy a quasilinear system of partial differential equations. Solutions are sought which only depend on a single function of the time and space coordinates.

Guy Boillat

Papers

Nonlinear Electrodynamics: Lagrangians and Equations of Motion

Hyperbolic Principal Subsystems: Entropy Convexity and Subcharacteristic Conditions

On the shock structure problem for hyperbolic system of balance laws and convex entropy

On the evolution law of weak discontinuities for hyperbolic quasi-linear systems

Simple Waves in N‐Dimensional Propagation