Existence result for a class of nonconservative and nonstrictly hyperbolic systems
Graziano Crasta Philippe G. LeFloch
Communications on Pure & Applied Analysis 2002, 1(4): 513-530 doi: 10.3934/cpaa.2002.1.513
We consider the class of nonconservative hyperbolic systems

$\partial_t u+A(u) \partial_x u =0,\quad\partial_t v + A(u) \partial_x v =0,$

where $u=u(x,t),\quad v=v(x,t)\in\mathbb R^N$ are the unknowns and $A(u)$ is a strictly hyperbolic matrix. Relying on the notion of weak solution proposed by Dal Maso, LeFloch, and Murat ("Definition and weak stability of nonconservative products", J. Math. Pures Appl. 74 (1995), 483--548), we establish the existence of weak solutions for the corresponding Cauchy problem, in the class of bounded functions with bounded variation. The main steps in our proof are as follows:
(i) We solve the Riemann problem based on a prescribed family of paths.
(ii) We derive a uniform bound on the total variation of corresponding wave-front tracking approximations $u^h$, $v^h$.
(iii) We justify rigorously the passage to the limit in the nonconservative product $A(u^h) \partial_x v^h$, based on the local uniform convergence properties of $u^h$, by extending an argument due to LeFloch and Liu ("Existence theory for nonlinear hyperbolic systems in nonconservative form", Forum Math. 5 (1993), 261--280). Our results provide a generalization to the existence theorem established earlier in the scalar case ($N=1$) by the second author ("An existence and uniqueness result for two nonstrictly hyperbolic systems", IMA Volumes in Math. and its Appl. 27, "Nonlinear evolution equations that change type", ed. B.L. Keyfitz and M. Shearer, Springer Verlag, 1990, pp. 126--138.)

keywords: nonconservative product weak solution wave-front tracking continuous dependence. function with bounded variation Nonlinear hyperbolic system
Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits
K. T. Joseph Philippe G. LeFloch
Communications on Pure & Applied Analysis 2002, 1(1): 51-76 doi: 10.3934/cpaa.2002.1.51
This paper is concerned with the boundary layers that arise in solutions of a nonlinear hyperbolic system of conservation laws in presence of vanishing diffusion. We consider self-similar solutions of the Riemann problem in a half-space, following a pioneering idea by Dafermos for the standard Riemann problem. The system is strictly hyperbolic but no assumption of genuine nonlinearity is made; moreover, the boundary is possibly characteristic, that is, the wave speed do not have a specific sign near the (stationary) boundary.
First, we generalize a technique due to Tzavaras and show that the boundary Riemann problem with diffusion admits a family of continuous solutions that remain uniformly bounded in the total variation norm. Careful estimates are necessary to cope with waves that collapse at the boundary and generate the boundary layer.
Second, we prove the convergence of these continuous solutions toward weak solutions of the Riemann problem when the diffusion parameter approaches zero. Following Dubois and LeFloch, we formulate the boundary condition in a weak form, based on a set of admissible boundary traces. Following Part I of this work, we identify and rigorously analyze the boundary set associated with the zero-diffusion method. In particular, our analysis fully justifies the use of the scaling $1/\varepsilon$ near the boundary (where $\varepsilon$ is the diffusion parameter), even in the characteristic case as advocated in Part I by the authors.
keywords: conservation law shock wave boundary layer vanishing diffusion self-similar solution.
A symmetrization of the relativistic Euler equations with several spatial variables
Philippe G. LeFloch Seiji Ukai
Kinetic & Related Models 2009, 2(2): 275-292 doi: 10.3934/krm.2009.2.275
We consider the Euler equations governing relativistic compressible fluids evolving in the Minkowski spacetime with several spatial variables. We propose a new symmetrization which makes sense for solutions containing vacuum states and, for instance, applies to the case of compactly supported solutions which are important to model star dynamics. Then, relying on these symmetrization and assuming that the velocity does not exceed some threshold and remains bounded away from the light speed, we deduce a local-in-time existence result for solutions containing vacuum states. We also observe that the support of compactly supported solutions does not expand as time evolves.
keywords: symmetric hyperbolic system vacuum relativistic fluid local well-posedness theory. Euler equations
$L^1$ continuous dependence for the Euler equations of compressible fluids dynamics
Paola Goatin Philippe G. LeFloch
Communications on Pure & Applied Analysis 2003, 2(1): 107-137 doi: 10.3934/cpaa.2003.2.107
We prove the $L^1$ continuous dependence of entropy solutions for the $2 \times 2$ (isentropic) and the $3\times 3$ (non-isentropic) systems of inviscid fluid dynamics in one-space dimension. We follow the approach developed by the second author for solutions with small total variation to general systems of conservation laws in [11, 14]. For the systems of fluid dynamics under consideration here, our estimates are more precise and we cover entropy solutions with large total variation.
keywords: large amplitude conservation law compressible fluids continuous dependence entropy solution Euler equations large total variation. uniqueness
Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity
Philippe G. Lefloch Cristinel Mardare Sorin Mardare
Discrete & Continuous Dynamical Systems - A 2009, 23(1&2): 341-365 doi: 10.3934/dcds.2009.23.341
Assuming minimal regularity assumptions on the data, we revisit the classical problem of finding isometric immersions into the Minkowski spacetime for hypersurfaces of a Lorentzian manifold. Our approach encompasses metrics having Sobolev regularity and Riemann curvature defined in the distributional sense, only. It applies to timelike, spacelike, or null hypersurfaces with arbitrary signature that possibly changes from point to point.
keywords: Lorentzian manifold Minkowski spacetime isometric embedding general hypersurface
Leaf superposition property for integer rectifiable currents
Luigi Ambrosio Gianluca Crippa Philippe G. Lefloch
Networks & Heterogeneous Media 2008, 3(1): 85-95 doi: 10.3934/nhm.2008.3.85
We consider the class of integer rectifiable currents without boundary in $\R^n\times\R$ satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.
keywords: Metric spaces valued $BV$ functions Multi-valued functions. Integer rectifiable currents Currents in metric spaces Cartesian currents

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