By Erwin Kreyszig
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For r ∈ L2 (Ω) we have r ∞ 2 0 (r, φRob )2 . k = k=1 For r ∈ H 1 (Ω) we get ν|r|21 ∞ ≤ a(r, r) = νμ2k (r, φRob )2 < ∞, k k=1 where the equal sign holds for r ∈ H01 (Ω). Furthermore, we have r ∈ H 2 (Ω) with (r − ϑ∂x r)(xL ) = 0 and (r + ϑ∂x r)(xR ) = 0, if and only if ν 2 |r|22 ∞ = (νμ2k )2 (r, φRob )2 < ∞. 11. 7, we have r Rob (t, ·) 2 0 ≤2 2 r0Rob 20 e−2νμ1 t 1 + 2 νμ1 t 0 t 2 ν|rRob (t, ·)|21 ≤ 2a(r0Rob , r0Rob )e−2νμ1 t + 0 Rob fLR (s, ·) Rob fLR (s, ·) 2 0 ds, 2 0 ds. 9), we have (r0Rob − ϑ∂x r0Rob )(xL ) = 0, Rob ∈ L2 ((0, T ); H 1 (Ω)) we get (r0Rob + ϑ∂x r0Rob )(xR ) = 0.
The weak maximum principle ensures uniqueness of the solution. It also allows to give statements on the positivity of the solutions. For non-negative initial data Di Di and rR and for a non-negative source term f , r0 , non-negative boundary data rL non-negativity of the solution follows by the weak maximum principle. In this case, the non-negative minimum is achieved on the parabolic boundary. 4. EXISTENCE AND REGULARITY OF THE SOLUTIONS 17 Neu Neu follows in the Neumann case, if we assume rL ≥ 0 and rR ≤ 0 (confer Ref.
Dt Upon multiplying the heat equation by t∂t r and integrating, the ensuing expression yields d xR ∂x r 20 + 2t ∂t r 20 − 2νt ∂t r∂x r xL ≤ 2t ∂t r 0 fLR 0 . νt dt t CHAPTER 3 The Goldstein-Taylor Model In this chapter, we study the velocity discrete system for our linear model problem. This a linear advection system, known as the Goldstein-Taylor model. The purpose of our investigation is to approximate the solutions of the heat equation by solutions of the advection system under consideration.