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The primitive equations on the large scale ocean under the small depth hypothesis
Interaction estimates and Glimm functional for general hyperbolic systems
1.  IACCNR, viale del Policlinico 137, 00161 ROMA 
$u_t + A(u) u_x = 0.\qquad\qquad (0.1)$
The aim of these estimates is to prove that there is Glimmtype functional $Q(u)$ such that
Tot.Var.$(u) + C_1 Q(u)$ is lower semicontinuous w.r.t. $L^1$ norm, $\qquad\qquad (0.2)$
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 (0.2).
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 (0.2) 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
Tot.Var.$(u)$ when
nonlinear waves of (0.1) interact, and the
introduction of a Glimm type functional $Q$, similar but not equivalent to Liu's
interaction functional [11].
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