Abstract
We consider a lattice regularization for an ill-posed diffusion equation with a trilinear constitutive law and study the dynamics of phase interfaces in the parabolic scaling limit. Our main result guarantees for a certain class of single-interface initial data that the lattice solutions satisfy asymptotically a free boundary problem with a hysteretic Stefan condition. The key challenge in the proof is to control the microscopic fluctuations that are inevitably produced by the backward diffusion when a particle passes the spinodal region.
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Barenblatt, G.I., Bertsch, M., Dal Passo, R., Ughi, M.: A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow. SIAM J. Math. Anal. 24(6), 1414–1439 (1993). https://doi.org/10.1137/0524082
Bellettini, G., Bertini, L., Mariani, M., Novaga, M.: Convergence of the One-dimensional Cahn-Hilliard equation. SIAM J. Math. Anal. 44(5), 3458–3480 (2012). https://doi.org/10.1137/120865410
Bellettini, G., Geldhauser, C., Novaga, M.: Convergence of a semidiscrete scheme for a forward-backward parabolic equation. Adv. Differ. Equ. 18(5/6), 495–522 (2013)
Bertsch, M., Smarrazzo, F., Tesei, A.: Nonuniqueness of solutions for a class of forward-backward parabolic equations. Nonlinear Anal. 137, 190–212 (2016). https://doi.org/10.1016/j.na.2015.12.028
Bonetti, E., Colli, P., Tomassetti, G.: A non-smooth regularization of a forward-backward parabolic equation. Math. Models Methods Appl. Sci. 27(4), 641–661 (2017). https://doi.org/10.1142/S0218202517500129
Braides, A.: Local Minimization, Variational Evolution and \({\varvec \Gamma}\) -Convergence. Lecture Notes in Mathematics, vol. 2094. Springer, Cham (2014) https://doi.org/10.1007/978-3-319-01982-6
Elliott, C.M.: The Stefan problem with a nonmonotone constitutive relation. IMA J. Appl. Math. 35(2), 257–264 (1985). https://doi.org/10.1093/imamat/35.2.257. Special issue: IMA conference on crystal growth (Oxford, 1985)
Esedoḡlu, S., Greer, J.B.: Upper bounds on the coarsening rate of discrete, ill-posed nonlinear diffusion equations. Commun. Pure Appl. Math. 62(1), 57–81 (2009). https://doi.org/10.1002/cpa.20259
Esedoḡlu, S., Slepcev, D.: Refined upper bounds on the coarsening rate of discrete, ill-posed diffusion equations. Nonlinearity 21(12), 2759–2776 (2008). https://doi.org/10.1088/0951-7715/21/12/002
Evans, L.C., Portilheiro, M.: Irreversibility and hysteresis for a forward-backward diffusion equation. Math. Models Methods Appl. Sci. 14(11), 1599–1620 (2004). https://doi.org/10.1142/S0218202504003763
Geldhauser, C., Novaga, M.: A semidiscrete scheme for a one-dimensional Cahn-Hilliard equation. Interfaces Free Bound. 13(3), 327–339 (2011). https://doi.org/10.4171/IFB/260
Gilding, B.H., Tesei, A.: The Riemann problem for a forward-backward parabolic equation. Physica D 239(6), 291–311 (2010). https://doi.org/10.1016/j.physd.2009.10.006
Gurevich, P., Shamin, R., Tikhomirov, S.: Reaction-diffusion equations with spatially distributed hysteresis. SIAM J. Math. Anal. 45(3), 1328–1355 (2013). https://doi.org/10.1137/120879889
Gurevich, P., Tikhomirov, S.: Rattling in spatially discrete diffusion equations with hysteresis (2016). ArXiv preprint no. arXiv:1601.05728
Helmers, M., Herrmann, M.: Interface dynamics in discrete forward-backward diffusion equations. Multiscale Model. Simul. 11(4), 1261–1297 (2013). https://doi.org/10.1137/130915959
Hilpert, M.: On uniqueness for evolution problems with hysteresis. Mathematical Models for Phase Change Problems (Óbidos. 1988), International Series of Numerical Mathematics, vol. 88, pp. 377–388. Birkhäuser, Basel (1989)
Holle, M.: Microstructure in Forward–Backward Lattice Diffusion. Master’s thesis, University of Bonn 2016
Höllig, K.: Existence of infinitely many solutions for a forward backward heat equation. Trans. Am. Math. Soc. 278(1), 299–316 (1983). https://doi.org/10.2307/1999317
Horstmann, D., Painter, K.J., Othmer, H.G.: Aggregation under local reinforcement: from lattice to continuum. Eur. J. Appl. Math. 15(5), 546–576 (2004). https://doi.org/10.1017/S0956792504005571
Lafitte, P., Mascia, C.: Numerical exploration of a forward–backward diffusion equation. Math. Models Methods Appl. Sci. 22(6), 1250,004, 33 (2012). https://doi.org/10.1142/S0218202512500042
Mascia, C., Terracina, A., Tesei, A.: Two-phase entropy solutions of a forward-backward parabolic equation. Arch. Ration. Mech. Anal. 194(3), 887–925 (2009). https://doi.org/10.1007/s00205-008-0185-6
Mielke, A., Truskinovsky, L.: From discrete visco-elasticity to continuum rate-independent plasticity: rigorous results. Arch. Ration. Mech. Anal. 203(2), 577–619 (2012). https://doi.org/10.1007/s00205-011-0460-9
Novick-Cohen, A., Pego, R.L.: Stable patterns in a viscous diffusion equation. Trans. Am. Math. Soc. 324(1), 331–351 (1991). https://doi.org/10.2307/2001511
Otto, F., Reznikoff, M.: Slow motion of gradient flows. J. Differ. Equ. 237(2), 372–420 (2007). https://doi.org/10.1016/j.jde.2007.03.007
Padrón, V.: Effect of aggregation on population revovery modeled by a forward–backward pseudoparabolic equation. Trans. Am. Math. Soc. 356(7), 2739–2756 (electronic) (2004). https://doi.org/10.1090/S0002-9947-03-03340-3
Peletier, M.A., Savaré, G., Veneroni, M.: Chemical reactions as \(\Gamma \)-limit of diffusion [revised reprint of mr2679596]. SIAM Rev. 54(2), 327–352 (2012). https://doi.org/10.1137/110858781
Perona, P., Malik, J.: Scale-space and edge-detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990). https://doi.org/10.1109/34.56205
Pierre, M.: Uniform convergence for a finite-element discretization of a viscous diffusion equation. IMA J. Numer. Anal. 30(2), 487–511 (2010). https://doi.org/10.1093/imanum/drn055
Plotnikov, P.I.: Passing to the limit with respect to viscosity in an equation with variable parabolicity direction. Differ. Equ. 30(4), 614–622 (1994)
Serfaty, S.: Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete Contin. Dyn. Syst. 31(4), 1427–1451 (2011). https://doi.org/10.3934/dcds.2011.31.1427
Smarrazzo, F., Tesei, A.: Long-time behavior of solutions to a class of forward-backward parabolic equations. SIAM J. Math. Anal. 42(3), 1046–1093 (2010). https://doi.org/10.1137/090763561
Smarrazzo, F., Tesei, A.: Some recent results concerning a class of forward–backward parabolic equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22(2), 175–188 (2011). https://doi.org/10.4171/RLM/594
Terracina, A.: Non-uniqueness results for entropy two-phase solutions of forward-backward parabolic problems with unstable phase. J. Math. Anal. Appl. 413(2), 963–975 (2014). https://doi.org/10.1016/j.jmaa.2013.12.045
Terracina, A.: Two-phase entropy solutions of forward-backward parabolic problems with unstable phase. Interfaces Free Bound. 17(3), 289–315 (2015). https://doi.org/10.4171/IFB/343
Visintin, A.: Quasilinear parabolic P.D.E.s with discontinuous hysteresis. Ann. Mat. Pura Appl. 185(4), 487–519 (2006). https://doi.org/10.1007/s10231-005-0164-6
Yin, J., Wang, C.: Young measure solutions of a class of forward-backward diffusion equations. J. Math. Anal. Appl. 279(2), 659–683 (2003). https://doi.org/10.1016/S0022-247X(03)00054-4
Zhang, K.: On existence of weak solutions for one-dimensional forward-backward diffusion equations. J. Differ. Equ. 220(2), 322–353 (2006). https://doi.org/10.1016/j.jde.2005.01.011
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Helmers, M., Herrmann, M. Hysteresis and Phase Transitions in a Lattice Regularization of an Ill-Posed Forward–Backward Diffusion Equation. Arch Rational Mech Anal 230, 231–275 (2018). https://doi.org/10.1007/s00205-018-1244-2
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DOI: https://doi.org/10.1007/s00205-018-1244-2