I am currently reading a paper on higher-order Cahn-Hilliard equations:
- Laurence Cherfils, Alain Miranville, Shuiran Peng, Higher-order anisotropic models in phase separation, Advances in Nonlinear Analysis, March 2017.
The problem states as follows:
\begin{align} \frac{\partial u}{\partial t} - \Delta \sum_{i=1}^k (-1)^i & \sum_{\lvert \alpha \rvert =i} a_{\alpha} \mathcal{D}^{2\alpha} u - \Delta F'(u) = 0\\ \mathcal{D}^{\alpha}u &= 0 \quad \text{ on } \partial\Omega, \quad \lvert \alpha \rvert \leq k, \\ u\bigl \lvert_{t=0} &= u_0 \end{align}
where
\begin{align} \mathcal{D}^{\alpha} = \frac{\partial^{\lvert \alpha \rvert}}{\partial_{x_1}^{k_1}\partial_{x_2}^{k_2}\partial_{x_3}^{k_3}} \end{align} with $\alpha = k_1+k_2+k_3 \in (\mathbb{N} \cup \{0\}^3)$ and $\lvert \alpha\rvert = k_1 + k_2 +k_3$.
In this paper, they provide a few a-priori estimates:
\begin{align} &\lVert u(t) \rVert^2_{H^k(\Omega)} \leq ce^{-c't}\bigl ( \lVert u_0 \rVert^2_{H^k(\Omega)} + \int_{\Omega} F(u_0) dx \bigr) + c'', \, \, c'>0, t \geq 0, \\ &\int_t^{t+r} \lVert (-\Delta)^{-\frac{1}{2}} \frac{\partial u}{\partial t} \rVert^2 \,ds \leq ce^{-c't} \bigl( \lVert u_0 \rVert^2_{H^k(\Omega)} + \int_{\Omega} F(u_0) dx \bigr) + c'', \quad c'>0, t\geq 0, r>0 \text{ given },\\ &\lVert u(t) \rVert_{H^{2k}(\Omega)} \leq Q (\lVert u_0 \rVert_{H^k(\Omega)} + c', \quad c>0, t\geq 1, \end{align}
Now they claim, that there exists a weak solution $u$ which (if we assume that $u_0 \in H^{k}_0(\Omega)$) has regularity \begin{align} u \in L^{\infty} (\mathbb{R}^+; H^k_0(\Omega)) \cap L^2(0,T; H^{2k}(\Omega) \cap H_0^k(\Omega))\\ \end{align} and
\begin{align} \frac{\partial u}{\partial t} \in L^2(0,T; H^{-1}(\Omega)). \end{align}
Regarding the proof, the authors only mention that the existence and regularity can be verified by using the above a-priori estimates and a standard Galerkin scheme.
I was trying to understand what is meant with "standard Galerkin scheme", but I wasn't very successful. Can someone please try to explain this approach to me?