## Pseudo-differential energy estimates of singular perturbations.(English)Zbl 0884.35183

The author considers hyperbolic nonlinear systems of the type $D_tu^d +A(u^d)u^d =0, \quad u^d(0) =u^d_0,\;d\to 0,$ where $$A=A_1+A_2$$ is a pseudo-differential operator; the symbol of $$A_1$$ is given by $$a_1(u^d(t,x),\xi)$$, depending on $$u^d$$; the operator $$A_2$$ is linear and translation invariant, with symbol of the form $$d^{-1} a_2(\xi)$$. By using a calculus of pseudo-differential operators with symbols of limited smoothness, the author obtains uniform bounds for the solution $$u^d$$ and studies the convergence of $$u^d$$ to the limit solution of the problem. Applications are given to singular limits arising in fluid mechanics and plasma physics.
Reviewer: L.Rodino (Torino)

### MSC:

 35S05 Pseudodifferential operators as generalizations of partial differential operators 35Q35 PDEs in connection with fluid mechanics 76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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