Narrow Results

We consider the Riemann problem for a Chapman–Jouguet combustion model which comes from Majda's model with a modified, bump-type ignition function proposed in [G. Lyng and K. Zumbrun, Arch. Rational Mech. Anal. 173 (2004) 213–277; Physica D 194 (2004) 1–29]. The unique Riemann solutions are obtained constructively under the pointwise and global entropy conditions. Furthermore, we prove rigorously that these solutions are the limits of the Riemann solutions for the corresponding self-similar Zeldovich–von Neumann–Döring model as the reaction rate goes to infinity. Finally we analyze the ignition problem for this Chapman–Jouguet combustion model, and the solutions show that the unburnt state is stable (respectively unstable) when the binding energy is small (respectively large), which is the desired property for a combustion model. We can also observe the phenomenon of the transition from a weak deflagration to a strong detonation which cannot occur for the Chapman–Jouguet combustion model corresponding to Majda's model with a step-type ignition function.

We study the large time behavior of entropic approximate solutions to one-dimensional, hyperbolic conservation laws with periodic initial data. Under mild assumptions on the numerical scheme, we prove the asymptotic convergence of the discrete solutions to a time- and space-periodic solution.

For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see [E. Yu. Panov, Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mat. Z.55(5) (1994) 116–129 (in Russian), Math. Notes55(5) (1994) 517–525]. This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Minimal entropy conditions for Burgers equation, Quart. Appl. Math.62(4) (2004) 687–700]. These two proofs both rely on the special connection between Hamilton–Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In this paper, we prove the single entropy result for scalar conservation laws without using Hamilton–Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case.

We study the initial value problem and the initial boundary value problem for nonlocal conservation laws. The nonlocal term is realized via a spatial integration of the solution between specified boundaries and affects the flux function of a given “local” conservation law in a multiplicative way. For a strictly convex flux function and strictly positive nonlocal impact we prove existence and uniqueness of weak entropy solutions relying on a fixed-point argument for the nonlocal term and an explicit Lax–Hopf-type solution formula for the corresponding Hamilton–Jacobi (HJ) equation. Using the developed theory for HJ equations, we obtain a semi-explicit Lax–Hopf-type formula for the solution of the corresponding nonlocal HJ equation and a semi-explicit Lax–Oleinik-type formula for the nonlocal conservation law.