By Carlisle D.P., Kaye R.
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For r0Rob ∈ H 2 (Ω) and fLR t 2 ν 2 |rRob (t, ·)|22 ≤ 2|r0Rob |22 e−2νμ1 t + 0 Rob Rob a(fLR , fLR )(s, ·) ds. 25) are valid. Estimates for higher order derivatives can be derived by applying similar procedures as described before. In the periodic case we seek for solutions in HP1 (Ω) := {u ∈ H 1 (Ω) : u(xL ) = u(xR )}. 13). A Hilbert basis in HP1 (Ω), consisting of orthogonal eigenP functions, is obtained by φP k k≥0 ∪ ψk k≥1 with φP k (x) := 2 cos |Ω| 2kπ (x − xL ) |Ω| for k = 0, 1, . . , ψkP (x) := 2 sin |Ω| 2kπ (x − xL ) |Ω| for k = 1, 2, .
1) are supplied by u (0, ·) = u0 (·) in Ω, v (0, ·) = v0 (·) in Ω. 1) are provided at the inflow boundaries, that is, u (·, xL ) = uL (·) in (0, T ), v (·, xR ) = vR (·) in (0, T ). In the sequel, we are interested in approximations of solutions of the heat equation. For this reason we use the data of the heat equation for the advection system. Let Di Di Neu Neu Rob Rob , rR rL , rR , rL and rR be the boundary data for the heat equation rL belonging to the Dirichlet, Neumann or Robin problem. The following choices for the boundary conditions are possible for the advection system.
In the Dirichlet case, compatibility conditions for the supplied data are determined by s0 (xL ) = sDi L (0), s0 (xR ) = sDi R (0), s1 (xL ) = ∂t sDi L (0), s1 (xR ) = ∂t sDi R (0) and Di 2 ν 2 ∂t2 sDi L (0) + ∂t sL (0) − ν∂x s0 (xL ) = h (0, xL ), Di 2 ν 2 ∂t2 sDi R (0) + ∂t sR (0) − ν∂x s0 (xR ) = h (0, xR ). Extensions are needed for the higher order derivatives. In the Neumann case we have to impose ∂x s0 (xL ) = sNeu L (0), ∂x s0 (xR ) = sNeu R (0), ∂x s1 (xL ) = ∂t sNeu L (0), ∂x s1 (xR ) = ∂t sNeu R (0) and Neu 3 ν 2 ∂t2 sNeu L (0) + ∂t sL (0) − ν∂x s0 (xL ) = ∂x h (0, xL ), Neu 3 ν 2 ∂t2 sNeu R (0) + ∂t sR (0) − ν∂x s0 (xR ) = ∂x h (0, xR ).