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)


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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