Suppose ${C}$ is solution of following problem:
$$\nabla\cdot \mathbf{u}=0\\\nabla\cdot(\mathbf{u}{C})-\nabla\cdot{(\mathbf{D}\nabla {C})}=0,\ {\mathbf{x}}\in \Omega, \\ {C}(\mathbf{x})=g(\mathbf{x}),\ \mathbf{x}\in \Gamma_{-},\\ (\mathbf{D}\nabla {C})\cdot \mathbf{n}=0,\ \mathbf{x}\in \Gamma_{n}\cup \Gamma_{+} $$ Here $\Gamma_{-}=\{\mathbf{x}\in \Gamma:\mathbf{u}\cdot \mathbf{n}<0\}$, $\Gamma_{+}=\{\mathbf{x}\in \Gamma:\mathbf{u}\cdot \mathbf{n}>0\}$,$\Gamma_{n}=\{\mathbf{x}\in \Gamma:\mathbf{u}\cdot \mathbf{n}=0\}$ I am trying to find stability bound of this problem. u is velocity here which is known constant, C is concenration of material of interest. Here $C\in H^1(\Omega)$.
My try: We do dot product with $C$ and perform integration by parts: $( \mathbf{u}\cdot \nabla C,C)= \frac{1}{2}\Big[\int_{\Gamma_{-}} ((g)^2)(\mathbf{u}\cdot \mathbf{n})ds\Big]+ \frac{1}{2}\Big[\int_{\Gamma_{+}} ((C)^2)(\mathbf{u}\cdot \mathbf{n})ds\Big]$ $-(\nabla\cdot{(\mathbf{D}\nabla{{C}})},{C})=-\int_{\Gamma}(\mathbf{D}\nabla{{C}})\cdot \mathbf{n}{C} \ ds+(\mathbf{D}\nabla{{C}},\nabla{{C}})=-\int_{\Gamma_{-}}(\mathbf{D}\nabla{{C}})\cdot \mathbf{n}g \ ds+(\mathbf{D}\nabla{{C}},\nabla{{C}})$
Then $$ \frac{1}{2}\Big[\int_{\Gamma_{+}} ((C)^2)(\mathbf{u}\cdot \mathbf{n})ds\Big]+(\mathbf{D}\nabla{{C}},\nabla{{C}})\leq\int_{\Gamma_{-}}(\mathbf{D}\nabla{{C}})\cdot \mathbf{n}g \ ds-\frac{1}{2}\Big[\int_{\Gamma_{-}} ((g)^2)(\mathbf{u}\cdot \mathbf{n})ds\Big]$$
How to handle first term in the left and side and first term in the right hand side of inequality now. I am aiming to have bound for L2 norm of $C$ and L2 norm of $\nabla C$