Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation

Abstract

We investigate totally linearly degenerate hyperbolic systems with relaxation. We aim to study their semilinear behavior, which means that the local smooth solutions cannot develop shocks, and the global existence is controlled by the supremum bound of the solution. In this paper we study two specific examples: the Suliciu-type and the Kerr-Debye-type models. For the Suliciu model, which arises from the numerical approximation of isentropic flows, the semilinear behavior is obtained using pointwise estimates of the gradient. For the Kerr-Debye systems, which arise in nonlinear optics, we show the semilinear behavior via energy methods. For the original Kerr-Debye model, thanks to the special form of the interaction terms, we can show the global existence of smooth solutions.

ODE singularity for blocked linearly degenerate quasilinear hyperbolic systems

“…We claim that when the strict hyperbolicity (12) is replaced by the constant multiplicity of characteristics (41) in the second block, we still have the same result, namely, we have Theorem 3.1 Under Assumptions (3), (13) and (41), the conclusion of Theorem 2.1 is still valid for the strictly block-hyperbolic system (6)- (8).…”

“…As solutions to ordinary differential equations and C 1 classical solutions to semilinear hyperbolic systems always have this kind of mechanism of singularity formation, it is called to be the 'ODE singularity' in [7--11] or the 'semilinear behavior' in [12,13].…”

“…All these results are obtained by the method of characteristics. On the other hand, by means of the energy method, the property of ODE singularity is proved in [12,13] for the Kerr-Debye model and its generalization.…”

On the Riemann Problem for a Hyperbolic System of Temple Class

“…In previous works, the classical Riemann problem for the Suliciu relaxation system was solved [5,9]. In this chapter, we show the unique entropy solution of the Riemann problem associated to the Suliciu relaxation system under assumptions of the conditions H1 and H2.…”

“…We refer to the system above as the Suliciu relaxation system [5,9,24] and correspond the case homogeneous with constant stress tensor of the model proposed by Bouchut and Boyaval, i.e., it is a simplified viscoelastic shallow fluid model. Also, this system can be considered as a relaxation for the isentropic Chaplygin gas dynamics system: ⎧ ⎨ ⎩ ρ t + (ρu) x = 0, (ρu) t + (ρu 2 + P ) x = 0, where ρ and u, respectively, stand for the density and the velocity of the gas, while the pressure P is given by the state equation…”