Subcritical Hamilton–Jacobi fractional equation in

Abstract

Solvability of Cauchy's problem in double-struckRN for fractional Hamilton–Jacobi equation (1.1) with subcritical nonlinearity is studied here both in the classical Sobolev spaces and in the locally uniform spaces. The first part of the paper is devoted to the global in time solvability of subcritical equation (1.1) in locally uniform phase space, a generalization of the standard Sobolev spaces. Subcritical growth of the nonlinear term with respect to the gradient is considered. We prove next the global in time solvability in classical Sobolev spaces, in Hilbert case. Regularization effect is used there to guarantee global in time extendibility of the local solution. Copyright © 2014 John Wiley & Sons, Ltd.

Navier–Stokes Equation and its Fractional Approximations

“…Consequently, for divergence-free f , an L 2 ( ) estimate is obtained: 10) thanks to the Poincaré inequality. Integrating the above, we obtain: 11) where c P denotes the constant in the Poincaré inequality.…”

“…But such technique offers further possible generalizations, first to study the problems, like e.g. Korteweg-de Vries equation and its extensions [7,8,12,13], where the solutions are obtained as a limit of solutions to parabolic regularizations of such equations (the method known as vanishing viscosity technique, originated by Hopf, Oleinik, Lax in 1950th); see also [9,10]. Another possible application of Henry's technique is to study critical problems (e.g.…”

New look at the Navier-Stokes equation

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