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I have the following diffusion advection with time and space dependent coefficients with Robin BC \begin{equation} \left \{ \begin{array}{l} \partial_t u - div(B(t,x) \nabla u) + V(t,x) \nabla u = 0 \\ B(t,0) \nabla u + V(t,0) u(t,0) = 0 \\ B(t,1) \nabla u + V(t,1) u(t,1) = g(t) \\ u(0,x) = u^0(x) \end{array} \right. \end{equation}

Space dimension d=1. regular domain $\Omega = (0,1)$, and finite time (0,T).

NB: The initial condition u^0 and the function g(t) are very regular $(C^\infty)$. The dissipation term $ B^i > 0$

1/ Under which assumptions ont the velocity $V$ this problem is well posed ?

2/ I would like to obtain some regularity on the (weak) solution (at least $C^1$ in space. Do you have an idea and/or a good reference to do that ??

Thank you very much for your help, i'm really blocked.

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1 Answer 1

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If we test the equation with $w\in H^1(0,1)$, integrating by parts yields: $$\int_0^1\partial_tuw+B(t,x)u_xw_x+Vu_xw=B(t,x)u_xw|^1_0$$ $$=(g(t)-V(t,1)u(t,1))w(t,1)+V(t,0)u(t,0)w(t,0)$$ for all $w\in H^1(0,1)$.

A step in the right direction for the equation being well posed is showing that the following bilinear form $$\mathcal{B}(u,w)=\int_0^1B(t,x)u_xw_x+Vu_xw,$$ is coercive. The assumption that $B>0$ and $V_x\le 0$ is sufficient here.

If I remember correctly, since the $V$ is involved in the Robin BC's, we also need $V\ge 0$.

Finally, for regularity theory, look at Evans' book on PDEs - chapter 7.

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