The weak solution to the Navier-Stokes equations in a bounded domain D ⊂ R3 with a smooth boundary is proved to be unique provided that it satisfies an additional requirement. This solution exists for all t ≥ 0. In a bounded domain D the solution decays exponentially fast as t → ∞ if the force term decays at a suitable rate.

Bending analysis of FGM plates under uniform and sinusoidal loaded result in forth order partial differential equation. In this paper the analytical solution is based on the extended Kantorovich iterative procedure. The differential equations for the iterative procedure is derived using the Galerkin method. The solution was develope based on the classical plate’s theory (CLPT). The reliability ...

This paper is devoted to the existence of a weak solution system describing self-propelled motion rigid body in viscous fluid whole ℝ 3 . The modelled by incompressible nonhomogeneous Navier-Stokes with nonnegative density. described balance linear and angular momentum. We consider case where slip allowed at fluid-solid interface through Navier condition prove global solution.

2-D and axisymmetric Navier-Stokes equations are solved using Reiman-Roe solver with different limiters for second-order accurate schemes. The results were obtained for supersonic viscous flows over semi-infinite axisymmetric and 2-D bodies. The free stream Mach numbers were 7.78 and 16.34. The stability of Roe method with different limiters and entropy conditions were considered. The results s...

We discuss the formulation of the Clay Mathematics Institute Millennium Prize Problem on the Navier-Stokes equations in the perspective of Hadamard’s notion of wellposedness. 1 The Clay Navier-Stokes Millennium Problem The Clay Mathematics Institute Millennium Prize Problem on the incompressible Navier-Stokes equations [3, 7] asks for a proof of (I) global existence of smooth solutions for all ...

We use the bivariate spline method to solve the steady state Navier-Stokes equations numerically. The bivariate spline we use in this paper is the space of splines of smoothness r and degree 3r over triangulated quadrangu-lations. The stream function formulation for the steady state Navier-Stokes equations is employed. Galerkin's method is applied to the resulting nonlinear fourth order equatio...

where u is a vector-valued function representing the velocity of the fluid, and p is the pressure. Note that the pressure depends in a non local way on the velocity u. It can be seen as a Lagrange multiplier associated to the incompressible condition (2). The initial value problem of the above equation is endowed with the condition that u(0, ·) = u0 ∈ L (R). Leray [11] and Hopf [6] had already ...