Abstract.We prove the small-time existence of a solution of the NavierStokes equations, for any initial data, for a free boundary fluid with surface tension taken into account. A fixed point method is used. The linearized problem is hyperbolic and dissipative. The classical methods to solve it seem to fail and the method used here could perhaps be applied for equations of the same kind.
We consider the solutions of the equation −ε 2 u + u − |u| p−1 u = 0 in S 1 × R, where ε and p are positive real numbers, p > 1. We prove that the set of the positive bounded solutions even in x 1 and x 2 , decreasing for x 1 ∈ ]−π, 0[ and tending to 0 as x 2 tends to +∞ is the first branch of solutions constructed by bifurcation from the ground-state solution (ε, w 0 ( x 2 ε )). We prove that there exists a positive real number ε such that for every ε ∈]0, ε ] there exists a finite number of solutions verifying the above properties and none such solution for ε > ε . The proves make use of compactness results and of the Leray-Schauder degree theory.
RésuméNous étudions l'équation −ε 2 u + u − |u| p−1 u = 0 dans S 1 × R, où ε et p sont des nombres réels strictement positifs, p > 1. Nous identifions l'ensemble des solutions (ε, u) où u est une fonction positive, paire en x 1 et x 2 , décroissante en x 1 dans [−π, 0] et tendant vers 0 quand x 2 tend vers +∞, comme la première branche de solutions issue d'une bifurcation à partir de l'état fondamental (ε, w 0 ( x 2 ε )). Nous prouvons qu'il existe un réel ε tel que pour tout ε ∈ ]0, ε ] il y a un nombre fini de solutions vérifiant les propriétés énoncées ci-dessus, et aucune telle solution pour ε > ε . Les preuves utilisent des résultats de compacité et la théorie du degré de Leray-Schauder.
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