On the multivariate Burgers equation and the incompressible Navier-Stokes equation (Part II)

Abstract

We consider global schemes with H2-estimates for the multivariate Burgers equation and the incompressible Navier-Stokes equation in its Leray projection form. We extend the definition of a global scheme and the global L2-theory of the multivariate Burgers equation to the incompressible Navier-Stokes equation, where we consider estimates of the gradient of the pressure in its Leray projection form in H1 at each time step of the scheme. The extended scheme has a simplified control function (simplified compared to [7]), which controls the growth of the Leray projection term. We use the fact that the data in our scheme are in H2 at each time step such that the data for the Poisson equation related to the elimination of pressure are in H1 at each approximation step. The upper bound of the growth of the velocity components is linear with respect to the time.

arXiv comment51 p., revised and extended, H^2 local contraction estimate of controlled scheme

References in corpus (6)

• Localisation and compactness properties of the Navier-Stokes global regularity problem
• Some infinite matrix analysis, a Trotter product formula for dissipative operators, and an algorithm for the incompressible Navier-Stokes equation
• Constructive analysis of the Navier-Stokes equation
• Global existence, regularity and a probabilistic scheme for a class of ultraparabolic Cauchy problems
• On local analytic expansions of densities in the context of (micro-)hypoelliptic and classes of semi-elliptic equations
• On a class of singular solutions to the incompressible 3-D Euler equation

Cited by in corpus (4)

• Some infinite matrix analysis, a Trotter product formula for dissipative operators, and an algorithm for the incompressible Navier-Stokes equation
• Constructive analysis of the Navier-Stokes equation
• On a class of singular solutions to the incompressible 3-D Euler equation
• On global schemes for highly degenerate Navier Stokes equation systems