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Abstract. Motivated by the "tug-of-war" game studied in [12], we consider a "non-local" version of the game which goes as follows: at every step two players pick respectively a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving of a fixed amount > 0 (as is done in the classical case), it is a s-stable Levy process which chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically we derive a deterministic non-local integro-differential equation that we call "infinity fractional Laplacian". We study existence, uniqueness, and regularity, both for the Dirichlet problem and for a double obstacle problem, both problems having a natural interpretation as "tug-of-war" games.

Abstract. We consider the steady-state Navier-Stokes equation in the whole space R 3 driven by a forcing function f . The class of source functions f under consideration yield the existence of at least one solution with finite Dirichlet integral ( ∇U 2 < ∞). Under the additional assumptions that f is absent of low modes and the ratio of f to viscosity is sufficiently small in a natural norm we construct solutions which have finite energy (finite L 2 norm). These solutions are unique among all solutions with finite energy and finite Dirichlet integral. The constructed solutions are also shown to be stable in the following sense: If U is such a solution then any viscous, incompressible flow in the whole space, driven by f and starting with finite energy, will return to U .

In this paper we address the existence, the asymptotic behavior and stability in L p and L p,∞ , 3 2 < p ≤ ∞, for solutions to the steady state 3D Navier-Stokes equations with possibly very singular external forces. We show that under certain smallness conditions of the forcing term there exists solutions to the stationary Navier-Stokes equations in L p spaces, and we prove the stability of these solutions. Namely, we prove that such small steady state solutions attract time dependent solutions driven by the same forcing, no matter how large the initial velocity of non-stationary solutions is.We also give non-existence results of stationary solutions in L p , for 1 ≤ p ≤ 3 2 .

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