Riv. Mat. Univ. Parma, Vol. 8, No. 1, 2017

Giovanni Naldi [a]

A journey through multiscale, some episodes from approximation and modelling

Pages: ( in press )
Received: 16 February 2017
Accepted in revised form: 25 July 2017
Mathematics Subject Classification (2010): 65M99, 35Q70, 35Q92, 65M06.
Keywords: Multiscale modelling, Relaxation approximation, nonlinear evolutionary differential equations, Relaxed methods.
Author address:
[a]: Università degli studi di Milano, Dipartimento di Scienze e Politiche Ambientali Centro ADAMSS, via Celoria 2, Milano, 20133, Italy

Abstract: The present notes contains both a survey of and some novelties about mathematical problems which emerged in multiscale based approach in approximation of evolutionary partial differential equations. Specifically, we present a relaxed systems approximation for nonlinear diffusion problems, which can tackle also the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation with a semi-linear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter \(\epsilon\). When \(\epsilon \rightarrow 0^+\), the system relaxes onto the original PDE: in this way, a consistent discretization of the relaxation system for vanishing \(\epsilon\) yields a consistent discretization of the original PDE. The advantage of this procedure is that numerical schemes obtained in this fashion do not require to solve implicit nonlinear problems and possess the robustness of upwind discretizations. We also review a unified framework, including BGK-based diffusive relaxation methods and new relaxed numerical schemes. A stability analysis for the new methods is sketched and high order extensions are provided. Finally some numerical tests in one and two dimensions are shown with preliminary results for nonlocal problems and multiscale hyperbolic systems.

This research was partially supported by ADAMSS Center of the Università degli Studi di Milano.

G. Aletti, A. Mentasti, and G. Naldi, From microscopic to macroscopic models for self-organized interacting populations, submitted.
P. Amorim, R. M. Colombo and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM Math. Model. Numer. Anal. 49 (2015), 19-37. MR3342191
D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws, SIAM J. Numer. Anal. 37 (2000), no. 6, 1973-2004. MR1766856
D. Aregba-Driollet, R. Natalini and S. Tang, Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems, Math. Comp. 73 (2004), no. 245, 63-94. MR2034111
D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749-1779. MR1885715
D. G. Aronson, Regularity properties of flows through porous media: A counterexample, SIAM J. Appl. Math. 19 (1970), 299-307. MR0265774
U. M. Ascher, S. J. Ruuth and R. J. Spiteri, Implicit-explicit Runge-Kutta methods for time dependent Partial Differential Equations, Appl. Numer. Math. 25 (1997), 151-167. MR1485812
U. M. Ascher, P. A. Markowich, P. Pietra and C. Schmeiser, A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model, Math. Models Methods Appl. Sci. 1 (1991), 347-376. MR1127572
G. I. Barenblatt, Scaling, self-similarity, and intermediate asymptotics, Cambridge Texts in Applied Mathematics, 14 , Cambridge University Press, Cambridge 1996. MR1426127
G. I. Barenblatt and J. L. Vázquez, Nonlinear diffusion and image contour enhancement, Interfaces Free Bound 6 (2004), no. 1, 31-54. MR2047072
A. E. Berger, H. Brézis and J. C. W Rogers, A numerical method for solving the problem \(u_t-\Delta f(u)=0\), RAIRO Anal. Numér. 13 (1979), 297-312. MR0555381
M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys. 125 (2006), no. 1, 249-280. MR2269565
S. Boscarino, F. Filbet and G. Russo, High order semi-implicit schemes for time dependent partial differential equations, J. Sci. Comput. 68 (2016), no. 3, 975-1001. MR3530996
S. Boscarino, P. G. LeFloch and G. Russo, High order asymptotic-preserving methods for fully nonlinear relaxation problems, SIAM J. Sci. Comput. 36 (2014), no. 2, A377-A395. MR3177364
S. Boscarino, L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comput. 35 (2013), no. 1, A22-A51. MR3033046
S. Boscarino and G. Russo, Flux-explicit IMEX Runge-Kutta schemes for hyperbolic to parabolic relaxation problems, SIAM J. Numer. Anal. 51 (2013), no. 1, 163-190. MR3033006
F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models, Numer. Math. 94 (2003), no. 4, 623-672. MR1990588
F. Bouchut, F. R. Guarguaglini and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations, Indiana Univ. Math. J. 49 (2000), no. 2, 723-749. MR1793689
F. Bouchut, H. Ounaissa and B. Perthame, Upwinding of the source term at interfaces for Euler equations with high friction, Comput. Math. Appl. 53 (2007), no. 3-4, 361-375. MR2323698
H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Functional Analysis 9 (1972), 63-74. MR0293452
T. Carleman, Problèmes Matématiques dans la théorie cinétique des gaz, Almqvist & Wiksells Boktryckeri Ab, Uppsala 1957. MR0098477
J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, J. Differential Equations 252 (2012), no. 4, 3245-3277. MR2871800
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in "Mathematical modeling of collective behavior in socio-economic and life sciences", G. Naldi, L. Pareschi and G. Toscani, eds., Model. Simul. Sci. Eng. Technol., Birkhäuser, Boston, MA 2010, 297-336. MR2744704
J. A. Carrillo, A. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys. 17 (2015), 233-258. MR3372289
F. Cavalli, Linearly implicit approximations of diffusive relaxation systems, Acta Appl. Math. 125 (2013), 79-103, MR3048641
F. Cavalli, Linearly implicit schemes for convection-diffusion equations, in "Hyperbolic problems: theory, numerics, applications", AIMS Ser. Appl. Math., 8, F. Ancona, A. Bressan, P. Marcati and A. Marson, eds., AIMS, Springfield, MO 2014, 423-430. MR3524348
F. Cavalli and G. Naldi, Time integration for semi-discrete approximation of multiscale hyperbolic systems, submitted.
F. Cavalli, G. Naldi, G. Puppo and M. Semplice, High-order relaxation schemes for non linear degenerate diffusion problems, SIAM J. Numer. Anal. 45 (2007), no. 5, 2098-2119. MR2346372
F. Cavalli, G. Naldi and M. Semplice, Parallel algorithms for nonlinear diffusion by using relaxation approximation, in "Numerical mathematics and advanced applications" (ENUMATH 2005 proc.), A. Bermúdez de Castro, D. Gómez, P. Quintela and P. Salgado, eds., Springer-Verlag, Berlin 2006, 404-411. MR2303667
F. Cavalli, G. Naldi, G. Puppo and M. Semplice, A family of relaxation schemes for nonlinear convection diffusion problems, Commun. Comput. Phys. 5 (2009), no. 2-4, 532-545. MR2513701
F. Cavalli, G. Naldi, G. Puppo and M. Semplice, Relaxed schemes based on diffusive relaxation for hyperbolic-parabolic problems: some new developments, in "Numerical methods for balance laws", Quad. Mat., 24, G. Puppo and G. Russo, eds., Seconda Univ. Napoli, Caserta 2010, 157-195. MR2976970
F. Cavalli, G. Naldi and I. Perugia, Discontinuous Galerkin approximation of relaxation models for linear and nonlinear diffusion equations, SIAM J. Sci. Comput. 34 (2012), no. 1, A105-A136. MR2890260
F. Cavalli, G. Naldi and I. Perugia, Discontinuous Galerkin approximation of porous Fisher-Kolmogorov equations, Commun. Appl. Ind. Math. 4 (2013), e-446, 18 pp. MR3130305
C. Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, 67, Springer-Verlag, New York 1988. MR1313028
S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, 3rd edition, Cambridge Univ. Press, London 1970. MR0258399
B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput. 16 (2001), no. 3, 173-261. MR1873283
R. M. Colombo, M. Garavello and M. Lécureux-Mercier, Non-local crowd dynamics, C. R. Math. Acad. Sci. Paris 349 (2011), 769-772. MR2825938
F. Coquel, M. Postel, N. Poussineau and Q. H. Tran, Multiresolution technique and explicit-implicit scheme for multicomponent flows, J. Numer. Math. 14 (2006), no. 3, 187-216. MR2288231
F. Cordier, P. Degond and A. Kumbaro, An asymptotic-preserving all-speed scheme for the Euler and Navier-Stokes equations, J. Comput. Phys. 231 (2012), no. 17, 5685-5704. MR2947989
F. Coron and B. Perthame, Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal. 28 (1991), no. 1, 26-42. MR1083323
M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265-298. MR0287357
P. Degond and M. Tang, All speed scheme for the low Mach number limit of the isentropic Euler equations, Commun. Comput. Phys. 10 (2011), no. 1, 1-31. MR2775032
S. Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, J. Comput. Phys. 229 (2010), no. 4, 978-1016. MR2576236
S. M. Deshpande, P. S. Kulkarni and A. K. Ghosh, New developments in kinetic schemes, Comput. Math. Appl. 35 (1998), no. 1-2, 75-93. MR1605139
G. Dimarco, L. Pareschi, and V. Rispoli, Asymptotically implicit schemes for the hyperbolic heat equation, in "Hyperbolic problems: theory, numerics, applications" (Proc. Conference HYP2012), AIMS Ser. Appl. Math., 8, AIMS, Springfield, MO 2014, 865-872. MR3524401
D. A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods, Series: Mathématiques & Applications, 69, Springer, Heidelberg 2012. MR2882148
R. Eftimie, Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review, J. Math. Biol. 65 (2012), no. 1, 35-75. MR2917194
S. Evje and K. H. Karlsen, Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations, Numer. Math. 83 (1999), 107-137. MR1702599
S. Evje and K. H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations, SIAM J. Numer. Anal. 37 (2000), no. 6, 1838-1860. MR1766850
W. E, Principles of multiscale modeling, Cambridge University Press, Cambridge 2011. MR2830582
B. Engquist, P. Lötstedt and O. Runborg, Multiscale methods in science and engineering, Lect. Notes Comput. Sci. Eng., 44, Springer-Verlag, Berlin 2005. MR2161967
R. C. Fetecau and W. Sun, First-order aggregation models and zero inertia limits, J. Differential Equations 259 (2015), no. 11, 6774-6802. MR3397338
F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys. 229 (2010), no. 20, 7625-7648. MR2674294
W. E. Fitzgibbon, The fluid dynamical limit of the Carleman equation with reflecting boundary, Nonlinear Anal. 6 (1982), no. 7, 695-702. MR0664146
A. Friedman, The Stefan problem in several space variables, Trans. Amer. Math. Soc. 133 (1968), 51-87. MR0227625
E. Gabetta and B. Perthame, Scaling limits for the Ruijgrok-Wu model of the Boltzmann equation, Math. Methods Appl. Sci. 24 (2001), no. 13, 949-967. MR1848250
E. Gabetta, L. Pareschi and G. Toscani, Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal. 34 (1997), no. 6, 2168-2194. MR1480374
S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math. 4 (1951), 129-156. MR0047963
L. Gosse, and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris 334 (2002), no. 4, 337-342. MR1891014
L. Gosse and N. Vauchelet, Numerical high-field limits in two-stream kinetic models and 1D aggregation equations, SIAM J. Sci. Comput. 38 (2016), no. 1, A412-A434. MR3461316
S. Gottlieb and C.-W. Shu, Total variation diminishing Runge-Kutta schemes, Math. Comp. 67 (1998), no. 221, 73-85. MR1443118
S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001), no. 1, 89-112. MR1854647
J. L. Graveleau and P. Jamet, A finite difference approach to some degenerate nonlinear parabolic equations, SIAM J. Appl. Math. 20 (1971), 199-223. MR0290600
J. Haack, S. Jin and J.-G. Liu, An all-speed asymptotic-preserving method for the isentropic Euler and Navier-Stokes equations, Commun. Comput. Phys. 12 (2012), no. 4, 955-980. MR2913446
W. Jäger and J. Kačur, Solution of porous medium type systems by linear approximation schemes, Numer. Math. 60 (1991), no. 3, 407-427. MR1137200
J. W. Jerome and M. E. Rose, Error estimates for the multidimensional two-phase Stefan problem, Math. Comp. 39 (1982), no. 160, 377-414. MR0669635
S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput. 21 (1999), no. 2, 441-454. MR1718639
S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Riv. Math. Univ. Parma (N.S.) 3 (2012), no. 2, 177-216. MR2964096
S. Jin, M. A. Katsoulakis and Z. Xin, Relaxation schemes for curvature-dependent front propagation, Comm. Pure Appl. Math. 52 (1999), 1587-1615. MR1711038
S. Jin and C. D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 126 (1996), 449-467. MR1404381
S. Jin and L. Pareschi, Asymptotic-preserving (AP) schemes for multiscale kinetic equations: a unified approach, in "Hyperbolic problems: theory, numerics, applications, Vol. I, II" (Magdeburg, 2000), Internat. Ser. Numer. Math., 140, 141, Birkhäuser, Basel 2001, 573-582. MR1882959
S. Jin, L. Pareschi and G. Toscani, Diffusive relaxation schemes for multiscale discrete velocity kinetic equations, SIAM J. Numer. Anal. 35 (1998), 2405-2439.
S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal. 38 (2000), no. 3, 913-936. MR1322811
S. Jin and Z. Xin, Numerical passage from systems of conservation laws to Hamilton-Jacobi equations, relaxation schemes, SIAM J. Numer. Anal. 35 (1998), no. 6, 2385-2404. MR1655852
H. G. Kaper, G. K. Leaf and S. Reich, Convergence of semigroups with an application to the Carleman equation, Math. Methods Appl. Sci. 2 (1980), no. 3, 303-308. MR0581208
J. Kačur, A. Handlovičová and M. Kačurová, Solution of nonlinear diffusion problems by linear approximation schemes, SIAM J. Numer. Anal. 30 (1993), no. 6, 1703-1722. MR1249039
C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math. 44 (2003), no. 1-2, 139-181. MR1951292
S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981), no. 4, 481-524. MR0615627
T. G. Kurtz, Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics, Trans. Amer. Math. Soc. 186 (1973), 259-272. MR0336482
C. Lattanzio and R. Natalini, Convergence of diffusive BGK approximations for nonlinear strongly parabolic systems, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 2, 341-358. MR1899825
R. J. LeVeque and M. Pelanti, A class of approximate Riemann solvers and their relation to relaxation schemes, J. Comput. Phys. 172 (2001), no. 2, 572-591. MR1857615
P. L. Lions and G. Toscani, Diffusive limit for finite velocity Boltzmann kinetic models, Rev. Mat. Iberoamericana 13 (1997), no. 3, 473-513. MR1617393
E. Magenes, Two-phase Stefan problems in several space variables (Italian), Matematiche (Catania) 36 (1981), no. 1, 65-108. MR0736797
E. Magenes, R. H. Nochetto and C. Verdi, Energy error estimates for a linear scheme to approximate nonlinear parabolic problems, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 655-678. MR0921832
P. Marcati and A. Milani, The one-dimensional Darcy's law as the limit of a compressible Euler flow, J. Differential Equations 84 (1990), no. 1, 129-147. MR1042662
P. Marcati and B. Rubino, Parabolic relaxation limit for hyperbolic systems of conservation laws, Rend. Circ. Mat. Palermo (2) Suppl. 45, part I (1996), 393-406. MR1461088
G. Martalò and G. Naldi, A model for the evacuation dynamics of a crowd in bounded domain with obstacles, submitted.
H. P. McKean, The central limit theorem for Carleman's equation, Israel J. Math. 21 (1975), 54-92. MR0423553
F. Miczek, F. K. Röpke and P. V. F. Edelmann, A new numerical solver for flows at various Mach numbers, Astronomy & Astrophysics 576 (2015), A50. DOI: 10.1051/0004-6361/201425059
D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: from individuals to populations, J. Math. Biol. 50 (2005), no. 1, 49-66. MR2117406
C.-D. Munz, S. Roller, R. Klein and K. J. Geratz, The extension of incompressible flow solvers to the weakly compressible regime, Comput. & Fluids 32 (2003), no. 2, 173-196. MR1966318
G. Naldi, A chemo-mechanical model for the single myofibril in striated muscle contraction, Meccanica (2017), DOI: 10.1007/s11012-017-0654-9.
G. Naldi and L. Pareschi, Numerical schemes for kinetic equations in diffusive regimes, Appl. Math. Lett. 11 (1998), no. 2, 29-35. MR1613061
G. Naldi and L. Pareschi, Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation, SIAM J. Numer. Anal. 37 (2000), no. 4, 1246-1270. MR1756424
G. Naldi, L. Pareschi and G. Toscani, Relaxation schemes for partial differential equations and applications to degenerate diffusion problems, Surveys Math. Indust. 10 (2002), no. 4, 315-343. MR2012453
G. Naldi and G. Russo, eds., Multiscale and adaptivity: modeling, numerics and applications, Lecture Notes in Math., 2040, CIME Found. Subser., Springer, Heidelberg; Fondazione CIME, Florence 2012. MR3075638
R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), no. 8, 795-823. MR1391756
R. H. Nochetto, A. Schmidt and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems, Math. Comp. 69 (2000), no. 229, 1-24. MR1648399
R. H. Nochetto and C. Verdi, Approximation of degenerate parabolic problems using numerical integration, SIAM J. Numer. Anal. 25 (1988), 784-814. MR0954786
A. Nonaka, A. S. Almgren, J. B. Bell, M. J. Lijewski, C. M. Malone and M. Zingale, MAESTRO: An adaptive low Mach number hydrodynamics algorithm for stellar flows, The Astrophysical Journal Supplement Series 188 (2010), no. 2, 358-383. DOI: 10.1088/0067-0049/188/2/358
L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput. 25 (2005), no. 1-2, 129-155. MR2231946
B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions, SIAM J. Numer. Anal. 29 (1992), no. 1, 1-19. MR1149081
B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal. 199 (2011), no. 3, 707-738. MR2771664
I. S. Pop and W.-A. Yong, A numerical approach to degenerate parabolic equations, Numer. Math. 92 (2002), no. 2, 357-381. MR1922924
A. Pulvirenti and G. Toscani, Fast diffusion as a limit of a two-velocity kinetic model, Rend. Circ. Mat. Palermo (2) Suppl. 45, part II (1996), 521-528. MR1461100
P. L. Roe and M. Arora, Characteristic-based schemes for dispersive waves. I. The method of characteristics for smooth solutions, Numer. Methods Partial Differential Equations 9 (1993), 459-505. MR1237356
F. Salvarani, Diffusion limits for the initial-boundary value problem of the Goldstein-Taylor model, Rend. Sem. Mat. Univ. Politec. Torino 57 (1999), no. 3, 209-220. MR1974553
F. Salvarani and G. Toscani, The diffusive limit of Carleman-type models in the range of very fast diffusion equations, J. Evol. Equ. 9 (2009), no. 1, 67-80. MR2501352
F. Salvarani and J. L. Vázquez, The diffusive limit for Carleman-type kinetic models, Nonlinearity 18 (2005), 1223-1248. MR2134892
M. Seaïd, High-resolution relaxation scheme for the two-dimensional Riemann problems in gas dynamics, Numer. Methods Partial Differential Equations 22 (2006), no. 2, 397-413. MR2201440
C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in "Advanced numerical approximation of nonlinear hyperbolic equations" (Cetraro, 1997), Lecture Notes in Math., 1697, CIME Found. Subser., Springer, Berlin 1998, 325-432. MR1728856
C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II, J. Comput. Phys. 83 (1989), no. 1, 32-78. MR1010162
R. Spigler and D. H. Zanette, A BGK model for chemical processes: the reaction-diffusion approximation, Math. Models Methods Appl. Sci. 4 (1994), 35-47. MR1259201
E. B. Tadmor and R. E. Miller, Modeling materials, Cambridge University Press, Cambridge UK, 2011.
G. I. Taylor, Diffusion by continuous movements, Proc. London Math. Soc. S2-20 (1922), 196-212. MR1577363
J. L. Vázquez, The Porous Medium Equation. Mathematical theory, Oxford University Press, Oxford 2007. MR2286292
J. L. Vázquez, Smoothing and decay estimates for nonlinear diffusion equations. Equations of porous medium type, Oxford University Press, Oxford 2006. MR2282669

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