I am here dealing with a problem of diffusion where the boundary of the domain is changing with time, and in particular I would like to show that the problem is well-posed in a class of solutions that is somewhat regular (like Hölder-continuous for example).
To make things clear, let me introduce some notations. We consider the moving boundary domain $Q := \cup_{t\in [0,T]} \{t\}\times \Omega_t \subset \mathbb{R}^{1+d}$ and we would like to have well-posedness of the equation :
$$ \left\{ \begin{array}{rll} u_t - \Delta u & = 0 & \text{in } Q\\ \nabla u.\mathbf{n} & = 0 & \text{on } \cup_{t\in [0,T]} \{t\}\times \partial \Omega_t\\ u(0,.) & = u_0 & \text{in } \Omega_0 \tag{HE-MB} \label{HE-MB} \end{array} \right. $$
Suppose that each domain $\Omega_t$ is given by a $C^1$-diffeo $y : [0,T]\times \Omega_0 \rightarrow Q$, that is $\Omega_t = y(t,\Omega_0)$ and we also suppose $\partial \Omega_t = y(t,\partial \Omega_0)$.
Supposing $u$ is a smooth solution of \ref{HE-MB} performing the change of variables $v(t,x) = u(t,y(t,x))$, we get the following problem with fixed boundary :
$$ \left\{ \begin{array}{rll} v_t & = \text{div} ([\nabla_x y]^{-1} \nabla_x v) + \mathbf{b}.\nabla_x v & \text{in }[0,T]\times \Omega\\ [[\nabla_x y]^{-1}\nabla_x v + \mathbf{b}v].\mathbf{n} & = 0 &\text{on }[0,T]\times \Omega_0\\ v(0,.) & = u_0 & \text{in } \Omega_0 \tag{HE-FB} \label{HE-FB} \end{array} \right. $$
This is an advection-diffusion equation, for which well-posedness can obtained if we assume that the drift $\mathbf{b}(t,x) = (\nabla_x y^T)^{-1}\partial_t y$ is divergence free. However $\mathbf{b}$ must have non-zero divergence if the volume of the spatial domain changes over time, so here my hopes fail. I only found references for advection-diffusion with non-zero divergence drift for problems posed on the whole space $\mathbb{R}^d$.
Another idea I had was to have the change of variables $\tilde{v}(t,x) = u(t,y(t,x))J(t,x)$ where $J(t,x) = |\det \nabla_x y(t,x)|$, so that the total variation of mass of $\tilde{v}$ would match that of $u$. But I couldn't find a nice form for the equation satisfied by $\tilde{v}$.
Any insights on the subject would be of great help. Thanks in advance.