We study the asymptotic time behavior of global smooth solutions to general entropy, dissipative, hyperbolic systems of balance laws in m space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach a constant equilibrium state in the L p -norm at a rate O(t -(m/2)(1-1/ p) ) as t -> ? for p ? [min{m, 2}, ?]. Moreover, we can show that we can approxi- mate, with a faster order of convergence, the conservative part of the solution in terms of the linearized hyperbolic operator for m >= 2, and by a parabolic equa- tion, in the spirit of Chapman-Enskog expansion in every space dimension. The main tool is given by a detailed analysis of the Green function for the linearized problem.

We consider the construction and the properties of the Riemann solver for the hyperbolic system \begin{equation}\label{E:hyp0} u_t + f(u)_x = 0, \end{equation} assuming only that $Df$ is strictly hyperbolic. In the first part we prove a general regularity theorem on the admissible curves $T_i$ of the $i$-family, depending on the number of inflection points of $f$: namely, if there is only one inflection point, $T_i$ is $C^{1,1}$. If the $i$-th eigenvalue of $Df$ is genuinely nonlinear, by it is well known that $T_i$ is $C^{2,1}$. However, we give an example of an admissible curve $T_i$ which is only Lipschitz continuous if $f$ has two inflection points. In the second part, we show a general method for constructing the curves $T_i$, and we prove a stability result for the solution to the Riemann problem. In particular we prove the uniqueness of the admissible curves for \eqref{E:hyp0}. Finally we apply the construction to various approximations to \eqref{E:hyp0}: vanishing viscosity, relaxation schemes and the semidiscrete upwind scheme. In particular, when the system is in conservation form, we obtain the existence of smooth travelling profiles for all small admissible jumps of \eqref{E:hyp0}.

We consider the problem of writing Glimm type interaction estimates for the hyperbolic system \begin{equation}\label{E:abs0} u_t + A(u) u_x = 0. \end{equation} %only assuming that $A(u)$ is strictly hyperbolic. The aim of these estimates is to prove that there is Glimm-type functional $Q(u)$ such that \begin{equation}\label{E:abs1} \TV(u) + C_1 Q(u) \ \text{is lower semicontinuous w.r.t.} \ L^1-\text{norm}, \end{equation} with $C_1$ sufficiently large, and $u$ with small BV norm. In the first part we analyze the more general case of quasilinear hyperbolic systems. We show that in general this result is not true if the system is not in conservation form: there are Riemann solvers, identified by selecting an entropic conditions on the jumps, which do not satisfy the Glimm interaction estimate \eqref{E:abs1}. Next we consider hyperbolic systems in conservation form, i.e. $A(u) = Df(u)$. In this case, there is only one entropic Riemann solver, and we prove that this particular Riemann solver satisfies \eqref{E:abs1} for a particular functional $Q$, which we construct explicitly. The main novelty here is that we suppose only the Jacobian matrix $Df(u)$ strictly hyperbolic, without any assumption on the number of inflection points of $f$. These results are achieved by an analysis of the growth of $\TV(u)$ when nonlinear waves of \eqref{E:abs0} interact, and the introduction of a Glimm type functional $Q$, similar but not equivalent to Liu's interaction functional \cite{liu:admis}.

As strictly hyperbolic system of conservation laws of the form $$ u_{t}+f(u)_x =0 , \quad u(0,x)=\bar u (x)$$ is considered, where $ u \in\bbfR^N$, $f:\bbfR^N \rightarrow\bbfR^N$ is smooth, especially from a numerical point of view, that means, a semidiscrete upwind scheme of this equation is investigated. If we suppose that the initial data $\bar u (x) $ of this problem have small total variation the author proves that the solution of the upwind scheme $$ {\partial u(t,x) \over \partial t} + { ( f(u(t,x))-f(u(t,x-\varepsilon))) \over \varepsilon} =0 $$ has uniformly bounded variation (BV) norm independent on $t$ and $\varepsilon$. Moreover the Lipschitz-continuous dependence of the solution of the upwind scheme $u^{\varepsilon}(t)$ on the initial data is proved. This solution $u^{\varepsilon}(t)$ converges in $ L_1$ to a weak solution of the corresponding hyperbolic system as $ \varepsilon \rightarrow 0$. This weak solution coincides with the trajectory of a Riemann semigroup which is uniquely determined by the extension of Liu's Riemann solver to general hyperbolic systems.

The paper concerns with a hyperbolic system of conservation laws in one space variable $$ u_t + f(u)_x = 0,\qquad u(0,x) = u_0(x), $$ where $ u \in \Bbb R^n$, $f:\Omega \subseteq \Bbb R^n \rightarrow \Bbb R^n.$ Let $ K_0 \subset \Omega $ be a compact and let $\delta_1 > 0 $ be sufficiently small such that $K_1 = \{ u \in \Bbb R^n: \text{dist}(u,K_0) \leq \delta_1\}\subset \Omega.$ \par Assuming that the Jacobian matrix $A = Df$ is uniformly strictly hyperbolic in $K_1, u_0(-\infty) \in K_0$ and that the total variation of $u_0$ is sufficiently small, then there exists a unique ``entropic" solution $u: [0,+\infty) \rightarrow BV(\Bbb R,\Bbb R^n).$