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June  2012, 5(2): 325-355. doi: 10.3934/krm.2012.5.325

## The Lifschitz-Slyozov equation with space-diffusion of monomers

 1 Project COFFEE, INRIA Sophia Antipolis Méditerranée Research Centre, & Labo. J. A. Dieudonné UMR 7351 CNRS{Université Nice Sophia Antipolis, Parc Valrose, 06108 Nice, France 2 Laboratoire de Mathématiques, UMR 8628, CNRS-Université Paris-Sud 11, Bât. 425, Faculté des Sciences d'Orsay, Université Paris-Sud 11, F-91405 Orsay cedex,, France 3 Project-Team SIMPAF, INRIA Lille Nord Europe Research Centre, Park Plazza, 40 avenue Halley, F-59650 Villeneuve d'Ascq cedex, France

Received  September 2011 Revised  January 2012 Published  April 2012

The Lifschitz--Slyozov system describes the dynamics of mass exchanges between macro--particles and monomers in the theory of coarsening. We consider a variant of the classical model where monomers are subject to space diffusion. We establish the existence--uniqueness of solutions for a wide class of relevant data and kinetic coefficients. We also derive a numerical scheme to simulate the behavior of the solutions.
Citation: Thierry Goudon, Frédéric Lagoutière, Léon M. Tine. The Lifschitz-Slyozov equation with space-diffusion of monomers. Kinetic & Related Models, 2012, 5 (2) : 325-355. doi: 10.3934/krm.2012.5.325
##### References:
 [1] H. Brézis, "Analyse Fonctionnelle. Théorie et Applications,", Collection Mathématiques Appliquées pour la Maîtrise, (1983). Google Scholar [2] J. A. Carrillo and T. Goudon, A numerical study on large-time asymptotics of the Lifschitz-Slyozov system,, J. Scient. Comp., 18 (2003), 429. Google Scholar [3] L. Châun-Hoàn, "Étude de la Classe des Opérateurs $m-$Accrétifs de $L^1(\Omega)$ et Accrétifs dans $L^\infty(\Omega)$,", Thèse de 3ème cycle, (1977). Google Scholar [4] M. K. Chen and P. W. Voorhees, The dynamics of transient Ostwald ripening,, Modelling Simul. Mater. Sci. Eng., 1 (1993), 591. doi: 10.1088/0965-0393/1/5/002. Google Scholar [5] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill Book Company, (1955). Google Scholar [6] J.-F. Collet and T. Goudon, Lifschitz-Slyozov equations: The model with encounters,, Transp. Theory Stat. Phys., 28 (1999), 545. doi: 10.1080/00411459908214517. Google Scholar [7] J.-F. Collet and T. Goudon, On solutions of the Lifschitz-Slyozov model,, Nonlinearity, 13 (2000), 1239. doi: 10.1088/0951-7715/13/4/314. Google Scholar [8] J.-F. Collet, T. Goudon, F. Poupaud and A. Vasseur, The Becker-Döring system and its Lifschitz-Slyozov limit,, SIAM J. Appl. Math., 62 (2002), 1488. doi: 10.1137/S0036139900378852. Google Scholar [9] J.-F. Collet, T. Goudon and A. Vasseur, Some remarks on the large-time asymptotic of the Lifschitz-Slyozov equations,, J. Stat. Phys., 108 (2002), 341. doi: 10.1023/A:1015404021853. Google Scholar [10] J. Conlon, On a diffusive version of the Lifschitz-Slyozov-Wagner equation,, J. Nonlinear Sc., 20 (2010), 463. doi: 10.1007/s00332-010-9065-y. Google Scholar [11] D. B. Dadyburjor and E. Ruckenstein, Kinetics of Ostwald ripening,, J. Crystal Growth, 40 (1977), 279. doi: 10.1016/0022-0248(77)90017-3. Google Scholar [12] C. Dellacherie and P.-A. Meyer, "Probabilités et Potentiel," chapitres I à IV,, Édition entièrement refondue, (1372). Google Scholar [13] R. Edwards, "Functional Analysis: Theory and Applications," Corrected reprint of the 1965 original,, Dover Publications, (1995). Google Scholar [14] T. Goudon, "Intégration: Intégrale de Lebesgue et Introduction à l'Analyse Fonctionnelle,", Ellipses, (2011). Google Scholar [15] S. Hariz and J.-F. Collet, A modified version of the Lifschitz-Slyozov model,, Applied Math. Lett., 12 (1999), 81. doi: 10.1016/S0893-9659(98)00138-4. Google Scholar [16] M. Herrmann, B. Niethammer and J. J. L. Velázquez, Self-similar solutions for the LSW model with with encounters,, J. Differential Equations, 247 (2009), 2282. Google Scholar [17] P. Laurençot, Weak solutions to the Lifschitz-Slyozov-Wagner equation,, Indiana Univ. Math. J., 50 (2001), 1319. Google Scholar [18] P. Laurençot, The Lifschitz-Slyozov equation with encounters,, Math. Models Methods Appl. Sci., 11 (2001), 731. doi: 10.1142/S0218202501001070. Google Scholar [19] P. Laurençot, The Lifschitz-Slyozov-Wagner equation with conserved total volume,, SIAM J. Math. Anal., 34 (2002), 257. doi: 10.1137/S0036141001387471. Google Scholar [20] I. M. Lifschitz and L. Pitaevski, "Cinétique Physique," Cours de Physique Théorique, Vol. 10, L. Landau-I. Lifschitz,, Mir, (1990). Google Scholar [21] I. M. Lifschitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions,, J. Phys. Chem. Solids, 19 (1961), 35. doi: 10.1016/0022-3697(61)90054-3. Google Scholar [22] B. Niethammer, A scaling limit of the Becker-Döring equations in the regime of small excess density,, J. Nonlinear Sci., 14 (2004), 453. doi: 10.1007/s00332-004-0638-5. Google Scholar [23] B. Niethammer and F. Otto, Ostwald ripening: The screening length revisited,, Calc. Var. Partial Differential Equations, 13 (2001), 33. Google Scholar [24] B. Niethammer and R. Pego, Non-self-similar behavior in the LSW theory of Ostwald ripening,, J. Stat. Phys., 95 (1999), 867. doi: 10.1023/A:1004546215920. Google Scholar [25] B. Niethammer and R. Pego, On the initial-value problem in the Lifschitz-Slyozov-Wagner theory of Ostwald ripening,, SIAM J. Math. Anal., 31 (2000), 467. doi: 10.1137/S0036141098338211. Google Scholar [26] B. Niethammer and R. Pego, The LSW model for domain coarsening: Asymptotic behavior for conserved total mass,, J. Stat. Phys., 104 (2001), 1113. doi: 10.1023/A:1010405812125. Google Scholar [27] B. Niethammer and R. L. Pego, Well-posedness for measure transport in a family of nonlocal domain coarsening models,, Indiana Univ. Math. J., 54 (2005), 499. doi: 10.1512/iumj.2005.54.2598. Google Scholar [28] B. Niethammer and J. J. L. Velázquez, Global well-posedness for an inhomogeneous LSW-model in unbounded domains,, Math. Ann., 328 (2004), 481. doi: 10.1007/s00208-003-0503-0. Google Scholar [29] B. Niethammer and J. J. L. Velázquez, On screening induced fluctuations in Ostwald ripening,, J. Stat. Phys., 130 (2008), 415. doi: 10.1007/s10955-007-9449-z. Google Scholar [30] O. Penrose, The Becker-Döring equations at large times and their connection with the LSW theory of coarsening,, J. Stat. Phys., 89 (1997), 305. doi: 10.1007/BF02770767. Google Scholar [31] T. Phillips, Trouble with Lifshitz, Slyozov and Wagner, Science News, NASA Science, 2003., Available from: \url{http://www.nasa.gov/vision/earth/technologies/coarsening_prt.htm}., (). Google Scholar [32] V. V. Sagalovich and V. V. Slyozov, Diffusive decomposition of solid solutions,, Sov. Phys. Usp., 30 (1987), 23. doi: 10.1070/PU1987v030n01ABEH002792. Google Scholar [33] J. Simon, Compact sets in $L^p(0,T;B)$,, Ann. Mat., 146 (1987), 65. Google Scholar [34] L. M. Tiné, T. Goudon and F. Lagoutière, Simulations of the Lifschitz-Slyozov equations with coagulations terms: Finite volumes schemes and anti-diffusive strategies,, preprint, (2011). Google Scholar [35] F. Trèves, "Topological Vector Spaces, Distributions and Kernels,", Unabridged republication of the 1967 original, (1967). Google Scholar

show all references

##### References:
 [1] H. Brézis, "Analyse Fonctionnelle. Théorie et Applications,", Collection Mathématiques Appliquées pour la Maîtrise, (1983). Google Scholar [2] J. A. Carrillo and T. Goudon, A numerical study on large-time asymptotics of the Lifschitz-Slyozov system,, J. Scient. Comp., 18 (2003), 429. Google Scholar [3] L. Châun-Hoàn, "Étude de la Classe des Opérateurs $m-$Accrétifs de $L^1(\Omega)$ et Accrétifs dans $L^\infty(\Omega)$,", Thèse de 3ème cycle, (1977). Google Scholar [4] M. K. Chen and P. W. Voorhees, The dynamics of transient Ostwald ripening,, Modelling Simul. Mater. Sci. Eng., 1 (1993), 591. doi: 10.1088/0965-0393/1/5/002. Google Scholar [5] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill Book Company, (1955). Google Scholar [6] J.-F. Collet and T. Goudon, Lifschitz-Slyozov equations: The model with encounters,, Transp. Theory Stat. Phys., 28 (1999), 545. doi: 10.1080/00411459908214517. Google Scholar [7] J.-F. Collet and T. Goudon, On solutions of the Lifschitz-Slyozov model,, Nonlinearity, 13 (2000), 1239. doi: 10.1088/0951-7715/13/4/314. Google Scholar [8] J.-F. Collet, T. Goudon, F. Poupaud and A. Vasseur, The Becker-Döring system and its Lifschitz-Slyozov limit,, SIAM J. Appl. Math., 62 (2002), 1488. doi: 10.1137/S0036139900378852. Google Scholar [9] J.-F. Collet, T. Goudon and A. Vasseur, Some remarks on the large-time asymptotic of the Lifschitz-Slyozov equations,, J. Stat. Phys., 108 (2002), 341. doi: 10.1023/A:1015404021853. Google Scholar [10] J. Conlon, On a diffusive version of the Lifschitz-Slyozov-Wagner equation,, J. Nonlinear Sc., 20 (2010), 463. doi: 10.1007/s00332-010-9065-y. Google Scholar [11] D. B. Dadyburjor and E. Ruckenstein, Kinetics of Ostwald ripening,, J. Crystal Growth, 40 (1977), 279. doi: 10.1016/0022-0248(77)90017-3. Google Scholar [12] C. Dellacherie and P.-A. Meyer, "Probabilités et Potentiel," chapitres I à IV,, Édition entièrement refondue, (1372). Google Scholar [13] R. Edwards, "Functional Analysis: Theory and Applications," Corrected reprint of the 1965 original,, Dover Publications, (1995). Google Scholar [14] T. Goudon, "Intégration: Intégrale de Lebesgue et Introduction à l'Analyse Fonctionnelle,", Ellipses, (2011). Google Scholar [15] S. Hariz and J.-F. Collet, A modified version of the Lifschitz-Slyozov model,, Applied Math. Lett., 12 (1999), 81. doi: 10.1016/S0893-9659(98)00138-4. Google Scholar [16] M. Herrmann, B. Niethammer and J. J. L. Velázquez, Self-similar solutions for the LSW model with with encounters,, J. Differential Equations, 247 (2009), 2282. Google Scholar [17] P. Laurençot, Weak solutions to the Lifschitz-Slyozov-Wagner equation,, Indiana Univ. Math. J., 50 (2001), 1319. Google Scholar [18] P. Laurençot, The Lifschitz-Slyozov equation with encounters,, Math. Models Methods Appl. Sci., 11 (2001), 731. doi: 10.1142/S0218202501001070. Google Scholar [19] P. Laurençot, The Lifschitz-Slyozov-Wagner equation with conserved total volume,, SIAM J. Math. Anal., 34 (2002), 257. doi: 10.1137/S0036141001387471. Google Scholar [20] I. M. Lifschitz and L. Pitaevski, "Cinétique Physique," Cours de Physique Théorique, Vol. 10, L. Landau-I. Lifschitz,, Mir, (1990). Google Scholar [21] I. M. Lifschitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions,, J. Phys. Chem. Solids, 19 (1961), 35. doi: 10.1016/0022-3697(61)90054-3. Google Scholar [22] B. Niethammer, A scaling limit of the Becker-Döring equations in the regime of small excess density,, J. Nonlinear Sci., 14 (2004), 453. doi: 10.1007/s00332-004-0638-5. Google Scholar [23] B. Niethammer and F. Otto, Ostwald ripening: The screening length revisited,, Calc. Var. Partial Differential Equations, 13 (2001), 33. Google Scholar [24] B. Niethammer and R. Pego, Non-self-similar behavior in the LSW theory of Ostwald ripening,, J. Stat. Phys., 95 (1999), 867. doi: 10.1023/A:1004546215920. Google Scholar [25] B. Niethammer and R. Pego, On the initial-value problem in the Lifschitz-Slyozov-Wagner theory of Ostwald ripening,, SIAM J. Math. Anal., 31 (2000), 467. doi: 10.1137/S0036141098338211. Google Scholar [26] B. Niethammer and R. Pego, The LSW model for domain coarsening: Asymptotic behavior for conserved total mass,, J. Stat. Phys., 104 (2001), 1113. doi: 10.1023/A:1010405812125. Google Scholar [27] B. Niethammer and R. L. Pego, Well-posedness for measure transport in a family of nonlocal domain coarsening models,, Indiana Univ. Math. J., 54 (2005), 499. doi: 10.1512/iumj.2005.54.2598. Google Scholar [28] B. Niethammer and J. J. L. Velázquez, Global well-posedness for an inhomogeneous LSW-model in unbounded domains,, Math. Ann., 328 (2004), 481. doi: 10.1007/s00208-003-0503-0. Google Scholar [29] B. Niethammer and J. J. L. Velázquez, On screening induced fluctuations in Ostwald ripening,, J. Stat. Phys., 130 (2008), 415. doi: 10.1007/s10955-007-9449-z. Google Scholar [30] O. Penrose, The Becker-Döring equations at large times and their connection with the LSW theory of coarsening,, J. Stat. Phys., 89 (1997), 305. doi: 10.1007/BF02770767. Google Scholar [31] T. Phillips, Trouble with Lifshitz, Slyozov and Wagner, Science News, NASA Science, 2003., Available from: \url{http://www.nasa.gov/vision/earth/technologies/coarsening_prt.htm}., (). Google Scholar [32] V. V. Sagalovich and V. V. Slyozov, Diffusive decomposition of solid solutions,, Sov. Phys. Usp., 30 (1987), 23. doi: 10.1070/PU1987v030n01ABEH002792. Google Scholar [33] J. Simon, Compact sets in $L^p(0,T;B)$,, Ann. Mat., 146 (1987), 65. Google Scholar [34] L. M. Tiné, T. Goudon and F. Lagoutière, Simulations of the Lifschitz-Slyozov equations with coagulations terms: Finite volumes schemes and anti-diffusive strategies,, preprint, (2011). Google Scholar [35] F. Trèves, "Topological Vector Spaces, Distributions and Kernels,", Unabridged republication of the 1967 original, (1967). Google Scholar
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