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Solutions fortes, solutions faibles d'équations aux dérivées partielles d'évolution.

Abstract : We present in the introduction classical properties of weak and strong solutions of partial differential equations. Chapter 2 is dedicated to the study of multipliers and paramultipliers between Sobolev spaces. If the pointwise multiplication operator by a function is bounded from a Sobolev space into another, we say that this function is a multiplier between these spaces. We define likewise paramultipliers by the boundedness of Bony's paraproduct operator. We prove an almost full description of multiplier and paramultiplier spaces. This description is applied in Chapter 3 to the study of the weak-strong uniqueness problem for the Navier-Stokes equation in dimension d > or = 3. It enables us to prove a weak-strong uniqueness theorem which generalizes most known results. We consider in Chapter 4 infinite energy solutions of the two-dimensional Navier-Stokes equation. A theorem of Gallagher and Planchon asserts that a global solution exists if the initial data belong to a critical Besov space ; we extend this result to the case where the initial data belong to @BMO, which seems optimal. We prove in Chapter 5 global existence results for the critical semi-linear wave equation (with polynomial non-linearity), for initial data of infinite energy and arbitrarily large norm. Two methods of non-linear interpolation are employed : the method of Calderon and the method of Bourgain ; they give complementary results. Some classical results are recalled in Chapter 6, and we mention in Chapter 7 some possible further developments.
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Pierre Germain. Solutions fortes, solutions faibles d'équations aux dérivées partielles d'évolution.. Equations aux dérivées partielles [math.AP]. Ecole Polytechnique X, 2005. Français. ⟨pastel-00001901⟩

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