Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation

<p> We consider the dynamics of a nonlinear partial differential equation perturbed by additive noise. Assuming that the underlying deterministic equation has an unstable equilibrium, we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium. More precisely, we show that most trajectories starting at the unstable equilibrium are driven away in two stages. After passing through a cylindrical region, most trajectories diverge from the deterministic equilibrium through a cone-shaped region which is centered around a finite-dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation, and on which the influence of the nonlinearity is surprisingly small. This abstract result is then applied to explain spinodal decomposition in the stochastic Cahn-Hilliard-Cook equation on a domain  <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This equation depends on a small interaction parameter  <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> ε </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\varepsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and one is generally interested in asymptotic results as  <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon right-arrow 0"> <mml:semantics> <mml:mrow> <mml:mi> ε </mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\varepsilon \to 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Specifically, we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach  <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon Superscript negative 2 plus dimension upper G slash 2"> <mml:semantics> <mml:msup> <mml:mi> ε </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>dim</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\varepsilon ^{-2+\dim G / 2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to the  <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H squared left-parenthesis upper G ...