Let us consider the transport PDE $$ u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon) $$ where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE reads $$ u^\epsilon_t + u^\epsilon_x= \frac{1}{\epsilon} u^\epsilon\big(1-(u^\epsilon)^2\big). $$
Does the limit of $u^\epsilon$ as $\epsilon \to 0$ exists? Can it be characterized?
