TY - JOUR T1 - On Short Wave-Long Wave Interactions in the Relativistic Context: Application to the Relativistic Euler Equations AU - Dias , João Paulo AU - Frid , Hermano JO - Communications in Mathematical Analysis and Applications VL - 3 SP - 289 EP - 303 PY - 2023 DA - 2023/09 SN - 2 DO - http://doi.org/10.4208/cmaa.2023-0005 UR - https://global-sci.org/intro/article_detail/cmaa/22015.html KW - Short wave-long wave interaction, Dirac equation, Thirring model, relativistic Euler equation. AB -

In this paper we introduce a model of relativistic short wave-long wave interaction where the short waves are described by the massless (1+3)-dimensional Thirring model of nonlinear Dirac equation and the long waves are described by the (1+3)-dimensional relativistic Euler equations. The interaction coupling terms are modeled by a potential proportional to the relativistic specific volume in the Dirac equation and an external force proportional to the square modulus of the Dirac wave function in the relativistic Euler equation. An important feature of the model is that the Dirac equations are based on the Lagrangian coordinates of the relativistic fluid flow. In particular, an important contribution of this paper is a clear formulation of the relativistic Lagrangian transformation. As far as the authors know the definition of the Lagrangian transformation given in this paper is new. Finally, we establish the short-time existence and uniqueness of a smooth solution of the Cauchy problem for the regularized model. This follows through the symmetrization of the relativistic Euler equation introduced by Makino and Ukai (1995) and requires a slight extension of a well known theorem of T. Kato (1975) on quasi-linear symmetric hyperbolic systems.