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A sequence of integro-differential equations approximating a viscous porous medium equation. (English) Zbl 0986.76088

The author considers a sequence of particular integro-differential equations, whose solutions \(\rho_N\) converge as \(N\) tends to infinity to the solution \(\rho\) of a viscous porous medium equation. It is demonstrated that, under suitable regularity conditions, the functions \(\rho_N\) are smooth uniformly in \(N\). Moreover, the author derives an asymptotic expansion for \(\rho_N\) as \(N\) tends to infinity. It precisely describes the convergence to \(\rho\). The results of this paper could be useful, in particular, for the numerical simulation of viscous porous medium equation by particle method.

MSC:

76S05 Flows in porous media; filtration; seepage
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
45K05 Integro-partial differential equations
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[1] Adams, R. A.: Sobolev Spaces. New York: Academic Press 1975. · Zbl 0314.46030
[2] Butt‘a, P. and J. L. Lebowitz: Hydrodynamic limit of Brownian particles interacting with short- and long-range forces. J. Stat. Phys. 94 (1999), 653 - 694. · Zbl 0948.60091 · doi:10.1023/A:1004593607665
[3] Caffarelli, L. A. and C. E. Kenig: Gradient estimates for variable coefficient parabolic equations and singular perturbation problems. Amer. J. Math. 120 (1998), 391 - 439. · Zbl 0907.35026 · doi:10.1353/ajm.1998.0009
[4] Fabes, E. B.: Gaussian upper bounds on fundamental solutions of parabolic equations; the method of Nash. Lect. Notes Math. 1563 (1993). · Zbl 0818.35036
[5] Friedman, A.: Partial Differential Equations of Parabolic Type. Malabar (Florida, USA): Robert E. Krieger Publ. Comp. 1983.
[6] Gärtner, J.: On the McKean-Vlasov limit for interacting diffusions. Math. Nachr. 137 (1988), 197 - 248. · Zbl 0678.60100 · doi:10.1002/mana.19881370116
[7] Kipnis, C. and C. Landim: Scaling Limits of Interacting Particle Systems. Berlin: Sprin- ger-Verlag 1999. · Zbl 0927.60002
[8] Lady\check zenskaja, O. A., Solonnikov, V. A. and N. N. Ural’ceva: Linear and Quasi-Linear Equations of Parabolic Type (Translations of Mathematical Monographs: Vol. 23). Prov- idence (R.I.): Amer. Math. Soc. 1968. · Zbl 0174.15403
[9] Oelschläger, K.: A law of large numbers for moderately interacting diffusion processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 69 (1985), 279 - 322. · Zbl 0549.60071 · doi:10.1007/BF02450284
[10] Oelschläger, K.: A fluctuation theorem for moderately interacting diffusion processes. Probab. Theory Rel. Fields 74 (1987), 591 - 616. · Zbl 0592.60064 · doi:10.1007/BF00363518
[11] Oelschläger, K.: On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes. Probab. Theory Rel. Fields 82 (1989), 565 - 586. · Zbl 0673.60110 · doi:10.1007/BF00341284
[12] Oelschläger, K.: On the connection between Hamiltonian many-particle systems and the hydrodynamical equations. Arch. Rat. Mech. Anal. 115 (1991), 297 - 310. 91 · Zbl 0850.70166 · doi:10.1007/BF00375277
[13] Oelschläger, K.: The description of many-particle systems by the equations for a viscous, compressible, barotropic fluid. Math. Models Methods Appl. Sci. 5 (1995), 887 - 922. · Zbl 0838.76002 · doi:10.1142/S0218202595000486
[14] Oelschläger, K.: An integro-differential equation modelling a Newtonian dynamics and its scaling limit. Arch. Rat. Mech. Anal. 137 (1997), 99 - 134. · Zbl 0880.70007 · doi:10.1007/s002050050024
[15] Oelschläger, K.: Simulation of the solution of a viscous porous medium equation by a particle method. Preprint 1998. · Zbl 1093.76053 · doi:10.1137/S0036142900363377
[16] Varadhan, S. R. S.: Scaling limits for interacting diffusions. Comm. Math. Phys. 135 (1991), 313 - 353. · Zbl 0725.60085 · doi:10.1007/BF02098046
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