# Special Solutions to a Nonlinear Coarsening Model with Local Interactions

## Abstract

We consider a class of mass transfer models on a one-dimensional lattice with nearest-neighbour interactions. The evolution is given by the backward parabolic equation \(\partial _t x = - \frac{\beta }{|\beta |} \Delta x^\beta \), with \(\beta \) in the fast diffusion regime \((-\infty ,0) \cup (0,1]\). Sites with mass zero are deleted from the system, which leads to a coarsening of the mass distribution. The rate of coarsening suggested by scaling is \(t^\frac{1}{1-\beta }\) if \(\beta \ne 1\) and exponential if \(\beta = 1\). We prove that such solutions actually exist by an analysis of the time-reversed evolution. In particular we establish positivity estimates and long-time equilibrium properties for discrete parabolic equations with initial data in \(\ell _+^\infty (\mathbb {Z})\).

## Keywords

Coarsening Infinite particle system Backward fast diffusion Discrete parabolic regularity## Mathematics Subject Classification

70F45 35K55 37L60## Notes

### Acknowledgements

The author would like to thank Barbara Niethammer and Juan J. L. Velázquez for inspiration, helpful discussions and proofreading. This work was supported by the German Research Foundation through the CRC 1060 *The Mathematics of Emergent Effects*.

## References

- Bonforte, M., Vazquez, J.L.: Global positivity estimates and Harnack inequalities for the fast diffusion equation. J. Funct. Anal.
**240**(2), 399–428 (2006)MathSciNetCrossRefGoogle Scholar - Esedoglu, 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)MathSciNetzbMATHGoogle Scholar - Esedoglu, S., Slepčev, D.: Refined upper bounds on the coarsening rate of discrete, ill-posed diffusion equations. Nonlinearity
**21**(12), 2759 (2008)MathSciNetCrossRefGoogle Scholar - Giacomin, G., Olla, S., Spohn, H.: Equilibrium Fluctuations for \(\nabla _{\varphi }\) Interface Model. Ann. Probab.
**29**(3), 1138–1172, 07 (2001)MathSciNetCrossRefGoogle Scholar - Glasner, K.B., Witelski, T.P.: Coarsening dynamics of dewetting films. Phys. Rev. E
**67**, 016302 (2003)CrossRefGoogle Scholar - Glasner, K.B., Witelski, T.P.: Collision versus collapse of droplets in coarsening of dewetting thin films. Phys. D Nonlinear Phenom.
**209**(1), 80–104 (2005). Non-linear dynamics of thin films and fluid interfacesMathSciNetCrossRefGoogle Scholar - Hellén, E.K.O., Krug, J.: Coarsening of sand ripples in mass transfer models. Phys. Rev. E
**66**, 011304 (2002)CrossRefGoogle Scholar - Henseler, R., Niethammer, B., Otto, F.: A reduced model for simulating grain growth. In: Colli, Pierluigi, Verdi, Claudio, Visintin, Augusto (eds.) Free Boundary Problems, pp. 177–187. Birkhäuser Basel, Basel (2004)Google Scholar
- Helmers, M., Niethammer, B., Velázquez, J.J.L.: Mathematical analysis of a coarsening model with local interactions. J. Nonlinear Sci.
**26**(5), 1227–1291 (2016)MathSciNetCrossRefGoogle Scholar - Kohn, R.V., Otto, F.: Upper bounds on coarsening rates. Commun. Math. Phys.
**229**(3), 375–395 (2002)MathSciNetCrossRefGoogle Scholar - Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math.
**80**(4), 931–954 (1958)MathSciNetCrossRefGoogle Scholar - van der Meer, R.M., van der Weele, J.P., Lohse, D.: Bifurcation diagram for compartmentalized granular gases. Phys. Rev. E Cover. Stat. Nonlinear Biol. Soft Matter Phys.
**63**(061304), 061304-1–061304-9 (2001)Google Scholar