# Check the well-posedness of the problem with level set equation

Given a domain $\Omega$ with 2 subdomains $\Omega_1,\Omega_2$ as figure below

Level set equation $$\partial_t\varphi + U\cdot\nabla\varphi =0$$

$U$ is velocity only presents inside $\Omega_1$. We assume $U$ is irrotational, i.e. $\nabla \times U=0$, which implies that the velocity ﬁeld can be derived from a potential function $\Phi$ defined in $\Omega$ such that $U=\nabla\Phi$. The potential equation is given by

$$\begin{cases} -\nabla\cdot(\nabla\Phi)&=-\beta_S \quad \text{in }\Omega_1\\ -\nabla\cdot(\nabla\Phi)&=0 \quad \text{in }\Omega_2 \end{cases}$$ with $\beta_S$ given.

I want to solve this level set equation, especially with unfitted mesh but I don't know the problem is well-posed or not.

• If $U$ is a vector, then $U=\nabla\cdot \Phi$ only makes sense if $\Phi$ is a matrix. But then it's not clear what that is supposed to mean for the two equations for $\Phi$. Can you clarify? Mar 28 '16 at 21:03
• @WolfgangBangerth: So sorry, it's a typo, $U=\nabla \Phi$ Mar 29 '16 at 7:13
• OK, better :-) What I'm still not clear about is what you mean with the question whether this problem is "well posed"? Do you mean have a unique solution? If so, in what space? And what do you know about boundary values for $\varphi$? Mar 29 '16 at 11:33
• Using only my intuition ;-), I would say the problem you gave can be for reasonable input data (the source $\beta_S$, boundary conditions) "well posed". The time plays here a role of parameter that gives in each time the definition for $\Omega_1$ and, consequently, the definition of source $\beta_S$. Similar problems are solved e.g. by immersed interface method and so on. To give a more useful answer, you may specify more precisely for which part of your model you have doubts about well posed form. Mar 30 '16 at 9:09