### A “natural” norm for the method of characteristics using discontinuous finite elements : 2D and 3D case

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We consider the numerical approximation of a first order stationary hyperbolic equation by the method of characteristics with pseudo time step k using discontinuous finite elements on a mesh ${\mathcal{T}}_{h}$. For this method, we exhibit a “natural” norm || ||h,k for which we show that the discrete variational problem ${P}_{h}^{k}$ is well posed and we obtain an error estimate. We show that when k goes to zero problem $\left({P}_{h}^{k}\right)$ (resp. the || ||h,k norm) has as a limit problem (Ph) (resp. the || ||h norm) associated to the...

We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation ${\partial}_{t}u+\frac{.}{\dot{}}\left(bu\right)=0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$. As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with bounded divergence....

Global existence results and long time behavior are provided for a mathematical model describing the propagation of Feline Panleucopenia Virus (FPLV) within a domestic cat population; two transmission modes are involved: a direct one from infective cats to susceptible ones, and an indirect one from the contaminated environment to susceptible cats. A more severe impact of the virus on young cats requires an age-structured model.

We study an example of the balance law with a multifunction source term, coming from the theory of granular media. We prove the existence of "weak entropy solutions" to this system, using the vanishing viscosity method and compensated compactness. Because of the occurrence of a multifunction we give a new definition of the weak entropy solutions.

We study the uniqueness and L¹-stability of the Cauchy problem for a 2 × 2 system coming from the theory of granular media [9,10]. We work in a class of weak entropy solutions. The appearance of a multifunction in a source term, given by the Coulomb-Mohr friction law, requires a modification of definition of the weak entropy solution [5,6].

Sufficient and necessary conditions for the existence and uniqueness of classical solutions to the Cauchy problem for the scalar conservation law are found in the class of discontinuous initial data and non-convex flux function. Regularity of rarefaction waves starting from discontinuous initial data and their dependence on the flux function are investigated and illustrated in a few examples.

We study oscillatory solutions of semilinear first order symmetric hyperbolic system $Lu=f(t,x,u,\stackrel{\u203e}{u})$, with real analytic $f$.The main advance in this paper is that it treats multidimensional problems with profiles that are almost periodic in $T,X$ with only the natural hypothesis of coherence.In the special case where $L$ has constant coefficients and the phases are linear, the solutions have asymptotic description$${u}^{\u03f5}=U(t,x,t/\u03f5,x/\u03f5)+o\left(1\right)$$where the profile $U(t,x,T,X)$ is almost periodic in $(T,X)$.The main novelty in the analysis is the space of profiles which...