Coercivity of the transport diffusion problem

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1 Coercivity of the transport-diffusion problem In the derivation of the coercivity of the transport diffusion problem the “troublesome” term is ∫ Ω (β∇v)v (1) Since the diffusion part, if you have a Dirichlet condition on a measurable portion of the boundary, gives ∫ Ω µ∇v·∇v = ∫ Ω µ|v∇|2≥ µ0 1 +C2 Ω ∥v∥2 H 1(Ω) (2) (whereµ0 is the lower bound of µ in Ω and CΩ the constant in the Poincar´ e inequality),it is sufficient to find the conditions by which the integral in (1) is not negative. Now, since (β∇v)v = ∑ j βj ∂v ∂xj v = 1 2 ∑ j βj ∂v2 ∂xj = 1 2β·∇v2 and β·∇v2 = div(βv2)−v2 divβ, we have ∫ Ω (β∇v)v = 1 2 ∫ Ω div(βv2)− 1 2 ∫ Ω v2 divβ = 1 2 ∫ ΓN v2β·n− 1 2 ∫ Ω v2 divβ. (3) We have used the divergence theorem and exploited the fact the v has zero trace on Γ D (it is zero on the Dirichlet portion of the boundary). So, sufficient conditions to have ∫ Ω(β∇v)v≥ 0, and thus coercivity, are divβ≤ 0 in Ω , β·n≥ 0 on Γ N. It means that it is safe to impose Neumann boundary conditions on the outflow boundary , i.e. where the flow is exiting the domain, or where it is tangent to the boundary, since in that caseβ·n≥ 0. It is unsafe to do otherwise, since the problem may become unstable (you may lose coercivity). So, in general, do not impose Neumann condition on the inflow boundary. There are techniques to stabilize Neumann conditions on the inflow boundary, but their study is beyond the scope of the course. Moreover they are normally applied to the Navier- Stokes equations, where often you do not know a priori if a Neumann boundary is inflow or outflow. The condition div β≤ 0 is sometimes less critical. Indeed in this case you can exploit part of the term in (2) by noting that (we assume that div β be bounded) ∫ Ω v2 divβ≤∥ divβ∥L∞(Ω)∥v∥2 L2(Ω)≤∥ divβ∥L∞(Ω)∥v∥2 H 1(Ω). and thus if divβ> 0…