Weak and strong solutions for a class of quasilinear Allen--Cahn systems

We consider a quasilinear Allen--Cahn system which arises when the gradient energy term in the Ginzburg--Landau energy also contains zero order terms.

Such systems offer significant advantages in applications, since surface tensions and mobilities can be easily calibrated.

The analysis for these systems is highly challenging, partly due to the fact that the gradient term in the energy is non-convex and since gradient terms appear quadratically in the weak formulation.

This explains why an existence theory has been lacking for nearly thirty years.

In this paper, we give the first existence and uniqueness results for such systems.

Firstly, we prove existence and uniqueness of local-in-time strong solutions using the theory of maximal regularity.

Here, non-standard techniques have to be applied due to the fact that linear constraints on the solution are involved and due to nonlinear boundary conditions.

Secondly, using a minimizing movement approach we show the existence of global-in-time weak solutions.

Here, the main difficulty arises from the fact that the underlying energy is not $\lambda$-convex.

We overcome this issue by proving higher integrability of the gradient of the solution, first showing that solutions are bounded and then using an approach by Giaquinta and Modica.

This finally allows us to pass to the limit in the time-discrete approximation.

Using the de Giorgi interpolation technique, we are also able to show a sharp energy decay property despite the lack of convexity of the energy.