We consider elliptic equations on R N +1 of the formwhere g(x, u) is a sufficiently regular function with g(•, 0) ≡ 0. We give sufficient conditions for the existence of solutions of (1) which are quasiperiodic in y and decaying as |x| → ∞ uniformly in y. Such solutions are found using a center manifold reduction and results from the KAM theory. We discuss several classes of nonlinearities g to which our results apply.

We consider the equation ∆u + uyy + f (x, u) = 0, (x, y) ∈ R N × R (1) where f is sufficiently regular, radially symmetric in x, and f (•, 0) ≡ 0. We give sufficient conditions for the existence of solutions of (1) which are quasiperiodic in y and decaying as |x| → ∞ uniformly in y. Such solutions are found using a center manifold reduction and results from the KAM theory. A required nondegeneracy condition is stated in terms of fu(x, 0) and fuu(x, 0), and is independent of higher-order terms in the Taylor expansion of f (x, •). In particular, our results apply to some quadratic nonlinearities.

No abstract

No abstract

We study the convergence of weak solutions of the Navier–Stokes equations with vanishing measurable viscous coefficients in domains with nonflat boundaries. Sufficient anisotropic conditions on the vanishing rates of the viscous coefficients are found to prove the convergence of Leray–Hopf weak solutions of the Navier–Stokes equations to solutions of the corresponding Euler equations. As the domains are not flat, we apply a change of variables to flatten the domains. We then construct explicit boundary layers for the system of Navier–Stokes equations in the upper‐half space with measurable viscous coefficients. The result is new even when the viscous coefficients are constant, and it recovers the classical results when domains are flat and with constant viscous coefficients.