Local existence and uniqueness of viscoelastic fluid flows in unbounded domains

In this Note, we present a result of local existence and uniqueness, for ar ty initial data, of the solutions to the equations of viscoelastic viscoela stic fluids of Jeffreys type (differential constitutive law). The system of equations is supposed to be verified in art unbounded domain Omega subset of R-N (N = 2 or 3), uniformly regular. The difficulty comes essentially fr om the loss of compactness in the case of unbounded domains. To overcome th is difficulty we use a local compactness method, which allows us to constru ct a sequence of solutions on subdomains Omega(n) which union covers Omega. After that, we pass to the limit to define a solution over the whole domai n. Finally we shaw the uniqueness of this solution in its class of regulari ty, by using an energy estimate. (C) Academie des Sciences/Elsevier, Paris.