Irreversibility and Hysteresis for a Forward–backward Diffusion Equation

Abstract

Abstract. Our intention in this paper is to publicize and extend somewhat important work of Plotnikov [P] on the asymptotic limits of solutions of viscous regularizations of an nonlinear diffusion PDE with a cubic nonlinearity. Since the formal limit PDE is in general ill-posed, we expect that the limit solves instead a corresponding diffusion equation with hysteresis effects. We employ entropy/entropy flux pairs to prove various assertions consistent with this expectation.

Long-time behaviour of two-phase solutions to a class of forward-backward parabolic equations

“…(ii) As already remarked, when u 0 ∈ L ∞ (Ω) is an arbitrary initial datum to problem (2.1) subject to the only assumption (H 2 ), passing to the limit as ε → 0 in the regularized problems (2.15) need not give a two-phase solution to problem (2.1) ( [EP,Pl1,ST]). In other words, global existence of two-phase solutions to both the Neumann initial-boundary value problem (2.1) and the Cauchy problem for equation (1.1) is proven to hold under assumption (H 3 ), but if we consider arbitrary initial data u 0 subject to the less restrictive assumption (H 2 ), the situation is more complicated and global existence actually remains an open problem.…”

“…( [EP,MTT1]). That is, jumps between the stable phases S 1 and S 2 occur only at the points (x, t) where the function v(x, t) takes the value A (jumps from S 2 to S 1 ) or B (jumps from S 1 to S 2 ).…”

“…For any ε > 0 the solution (u ε , v ε ) to problem (2.15)-(2.16) satisfies a family of viscous entropy inequalities, this terminology being suggested by a formal analogy with the entropy inequalities for viscous conservation laws ( [EP,MTT2,NP,Pl1]). PROPOSITION 2.7 For any u 0 ∈ L ∞ (Ω) and ε > 0 let (u ε , v ε ) be the solution of problem (2.15)-(2.16) given by Theorem 2.6.…”

“…As already remarked, crucial consequences of the viscous entropy inequalities (2.18) are a priori estimates for both the families {u ε } and {v ε }. In particular in [Pl1] (see also [EP,MTT1,ST]) the following estimates are proved:…”

“…In other words, for arbitrary initial data u 0 ∈ L ∞ (Ω), the limiting function u is a superposition of different phases and the coefficients λ i (i = 0, 1, 2) can be regarded as phase fractions (see also [EP,MTT1,Sm,ST]). In the light of the above characterization, a natural question is whether all the coefficients λ i play a role in (2.28) or whether under suitable assumptions on the initial datum u 0 we can arrange that v = φ(u) almost everywhere in Q.…”