Ben Moussa, B.; Kossioris, G. T. On the system of Hamilton-Jacobi and transport equations arising in geometrical optics. (English) Zbl 1029.78002 Commun. Partial Differ. Equations 28, No. 5-6, 1085-1111 (2003). The paper under review deals with systems of Hamilton-Jacobi equation of eikonal type coupled with a linear transport equation of the form \[ \begin{cases} \partial_t\varphi+H(t,x, \nabla \varphi)=0 & \varphi(0, x)=\varphi_0(x)\\ \partial_t \sigma +\text{div}(f(\nabla \varphi)\sigma)=0 & \sigma(0, x)=\sigma_0(x) \end{cases}\quad x\in {\mathbb R}^d\tag{1} \] One looks first for viscosity solutions [see M.G. Crandall, H. Ishii and {P.-L. Lions}, Bull. Am. Math. Soc., New Ser. 27, 1–67 (1992; Zbl 0755.35015)] of the eikonal equation from (1). However, the transport equation is more difficult to handle, since \(\nabla \varphi\) may be discontinuous and therefore \(f(\nabla \varphi) \sigma\) is not a priori well defined. To solve this problem, the authors chose to use the notion of a measure solution given by F. Poupaud and M. Rascle in [Commun. Partial Differ. Equations 22, 337–358 (1997; Zbl 0882.35026)]. If \(T>0\) is given, one establishes that, under certain conditions on \(H\) (among them, a strict convexity condition) and \(f\), the system (1) has a unique couple of viscosity-measure solution \((\varphi, \sigma)\in W^{1,\infty}([0,T]\times {\mathbb R}^d)\times C([0,T], {\mathcal M}({\mathbb R}^d)\), provided the initial data \((\varphi_0, \sigma_0)\) belongs to \(W^{1, \infty}({\mathbb R}^d)\times {\mathcal M}({\mathbb R}^d)\). This solution is stable under approximating the eikonal equation by a parabolic one (the vanishing viscosity approximation), or by regularizing \(\varphi\) by a convolution in the transport equation. The above-mentioned conditions on \(H\) and \(f\) turn out to be satisfied for certain systems obtained when approximating high frequency wave fields in the following cases: Schrödinger equation, spinless Bethe-Salpeter equation and Helmholtz equation. The remaining results of the paper are devoted to the study of the geometric structure of the gradient field \(\nabla \varphi\) and the properties of the measure solutions \(\sigma\) for the general system that appears in the high frequency approximation for the Schrödinger and Helmholtz equation. One first discusses the fact that, unlike in the classical case, the Filippov characteristics are not necessarily optimal trajectories. Then, one studies the geometry of the measure solution of the transport equation around shock waves of the measure solution of the eikonal equation. Reviewer: Ingrid Beltita (Bucureşti) Cited in 5 Documents MSC: 78A05 Geometric optics 35F25 Initial value problems for nonlinear first-order PDEs 78A40 Waves and radiation in optics and electromagnetic theory 35Q60 PDEs in connection with optics and electromagnetic theory 35L67 Shocks and singularities for hyperbolic equations Keywords:eikonal equation; transport equation; viscosity solution; measure solution; Filippov characteristics; high frequency characteristics Citations:Zbl 0882.35026; Zbl 0755.35015 PDFBibTeX XMLCite \textit{B. Ben Moussa} and \textit{G. T. Kossioris}, Commun. Partial Differ. Equations 28, No. 5--6, 1085--1111 (2003; Zbl 1029.78002) Full Text: DOI References: [1] DOI: 10.1007/s002050100176 · Zbl 1043.35052 [2] Aubin J. P., A Series of Comprehensive Studies in Mathematics pp 264– (1984) [3] Barles G., Mathématiques & Applications pp 17– (1994) [4] DOI: 10.1137/S0036142996307119 · Zbl 0949.65095 [5] Ben Moussa, B. 1999. 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