We summarize a recently developed integral equation approach to tackling the long-time existence problem for smooth solution v(x, t) to the 3-D Navier-Stokes equation in the context of a periodic box problem with smooth time-independent forcing and initial condition v0. Using an inverse Laplace transform of v̂(k, t) − v̂0 in 1/t, we arrive at an integral equation for Û(k, p), where p is inverse-L...

This paper deals with the existence of solutions a p-biharmonic pseudo parabolic partial differential equation logarithmic nonlinearity in bounded domain. We prove global weak using Faedo-Galerkin method and applying concavity approach, that blow up at finite time. Further, we provide maximal limit for blow-up

We consider the interior Hölder regularity of spatial gradient viscosity solution to parabolic normalized p ( x , t ) -Laplace equation u = ? i j + ? 2 | D with some suitable assumptions on which arises naturally from a two-player zero-sum stochastic differential game probabilities depending space and time.

We present two methods for solving the electrostatics of point charges and multipoles on the surface of a sphere, i.e., in the space S2, with applications to numerical simulations of two-dimensional (2D) polar fluids. In the first approach, point charges are associated with uniform neutralizing backgrounds to form neutral pseudo-charges, while in the second, one instead considers bi-charges, i....