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We prove global existence of smooth solutions of the 3D log-log energy-supercritical wave equation
with and smooth initial data . First we control the norm of the solution on an arbitrary size time interval by an expression depending on the energy and an a priori upper bound of its norm, with . The proof of this long time estimate relies upon the use of some potential decay estimates and a modification of an argument by Tao. Then we find an a posteriori upper bound of the norm of the solution by combining the long time estimate with an induction on time of the Strichartz estimates.
We prove the local well-posedness of the periodic stochastic Korteweg–de Vries equation with the additive space-time white noise. To treat low regularity of the white noise in space, we consider the Cauchy problem in the Besov-type space for , such that . In establishing local well-posedness, we use a variant of the Bourgain space adapted to and establish a nonlinear estimate on the second iteration on the integral formulation. The deterministic part of the nonlinear estimate also yields the local well-posedness of the deterministic KdV in , the space of finite Borel measures on .
Let . We consider the global Cauchy problem for the generalized Navier–Stokes system
for and , where is smooth and divergence free, and is a Fourier multiplier whose symbol is nonnegative; the case is essentially Navier–Stokes. It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes for . We improve this slightly by establishing global regularity under the slightly weaker condition that for all sufficiently large and some nondecreasing function such that . In particular, the results apply for the logarithmically supercritical dissipation .