Weak Solutions and Convergent Numerical Schemes of Brenner-navier-stokes Equations

Lately, there has been some interest in modifications of the compressible Navier-Stokes equations to include diffusion of mass. In this paper, we investigate possible ways to add mass diffusion to the 1-D Navier-Stokes equations without violating the basic entropy inequality. As a result, we recover a general form of Brenner’s modification of the Navier-Stokes equations. We consider Brenner’s system along with another modification where the viscous terms collapse to a Laplacian diffusion. For each of the two modifications, we derive a priori estimates for the PDE, sufficiently strong to admit a weak solution; we propose a numerical scheme and demonstrate that it satisfies the same a priori estimates. For both modifications, we then demonstrate that the numerical schemes generate solutions that converge to a weak solution (up to a subsequence) as the grid is refined. 1. Conservation laws Consider the system of conservation laws in one space dimension: ut + f(u)x = 0, x ∈ Ω, 0 ≤ t ≤ T (1) u(x, 0) = u(x), Here u = (u1, ..., un) > is the vector of unknowns and the fluxes f = (f1, f2, · · · , fn) are Lipschitz continuous functions of u. Ω is a bounded domain in one dimension (1-D). (We take Ω = (0, 1).) The system is also subject to appropriate boundary conditions. T is an arbitrary finite time. u(x) is a suitably bounded initial datum. Conservation laws are often endowed with entropies. Entropy is a useful tool to obtain a priori bounds on the solution and sometimes infer uniqueness. We will briefly introduce the concept. Let (U,F ) denote an entropy and entropy flux (for short, entropy pair). By definition U u fu = Fu, and Uu = w T is termed the entropy variables. Using the entropy variables, (1) can be rewritten as uwwt + g(w)x = 0, x ∈ Ω (2) where uw is symmetric and positive definite and gw is symmetric. (See [Moc80]). Often the conservation law is considered to be a model of an associated viscous equation, ut + f(u)x = (G(u)ux)x, x ∈ Ω (3) u(x, 0) = u(x), where G(u) is a matrix. The regularization (G(u)ux) is conservative and we will refer to (3) as being conservative. Using the entropy variables, (3) can be stated as uwwt + g(w)x = (G̃(w)wx)x, x ∈ Ω. (4) We require that G̃ is symmetric and positive semi-definite. (This property ensures that entropy is diffused.) Note that G̃wx = Gux = F V where F is commonly known as the viscous flux. Date: January 20, 2015. 1