Application: The Cahn—Hilliard Equation

In this chapter we consider the equation(17.1)$$\frac{\partial }{{\partial t}}u + \frac{{{\partial ^4}}}{{\partial {x^4}}}u + \frac{{{\partial ^2}}}{{\partial {x^2}}}p\left( u \right) = 0$$ where(17.2)$$p\left( {u = - {b_2}} \right){L^{ - 2}}u - {b_3}{L^{ - 1}}{u^2} - {b_4}{u^3}$$ on the space H = {u∈ L2(0,L): ∫₀Ludx = 0} (the same as in Chapter 16) with the periodic boundary conditions. This means in particular that A = (d4/dx4)u and N(u) = Au − A1/2p(u), for u ∈ H4(0,L) (= the L2-Sobolev space of order 4) such that(17.3)$$ \begin{gathered} u\left( 0 \right) = u\left( L \right),{\text{ }}u'\left( 0 \right) = u'\left( L \right),{\text{ }}u''\left( 0 \right) = u\left( L \right), \hfill \\ {\text{ }}u'''\left( 0 \right) = u'''\left( L \right) \hfill \\ \end{gathered} $$