## Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems

HTML articles powered by AMS MathViewer

- by Donatella Donatelli and Pierangelo Marcati PDF
- Trans. Amer. Math. Soc.
**356**(2004), 2093-2121 Request permission

## Abstract:

In this paper we investigate the diffusive zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: \[ W_{t}(x,t) + \frac {1}{\varepsilon }A(x,D)W(x,t)= \frac {1}{\varepsilon ^2} B(x,W(x,t))+\frac {1}{\varepsilon } D(W(x,t))+E(W(x,t)).\] We analyze the singular convergence, as $\varepsilon \downarrow 0$, in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps:

[(i)] We single out algebraic “structure conditions” on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories.

[(ii)] We deduce “energy estimates ”, uniformly in $\varepsilon$, by assuming the existence of a symmetrizer having the so-called block structure and by assuming “dissipativity conditions” on $B$.

[(iii)] We assume a Kawashima type condition and perform the convergence analysis by using generalizations of compensated compactness due to Tartar and Gérard.

Finally, we include examples that show how to use our theory to approximate any quasilinear parabolic systems, satisfying the Petrowski parabolicity condition, or general reaction diffusion systems, including Chemotaxis and Brusselator type systems.

## References

- 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. MR**1793689**, DOI 10.1512/iumj.2000.49.1811 - Jacques Chazarain and Alain Piriou,
*Introduction to the theory of linear partial differential equations*, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR**678605** - Gui Qiang Chen, C. David Levermore, and Tai-Ping Liu,
*Hyperbolic conservation laws with stiff relaxation terms and entropy*, Comm. Pure Appl. Math.**47**(1994), no. 6, 787–830. MR**1280989**, DOI 10.1002/cpa.3160470602 - Bernard Dacorogna,
*Weak continuity and weak lower semicontinuity of nonlinear functionals*, Lecture Notes in Mathematics, vol. 922, Springer-Verlag, Berlin-New York, 1982. MR**658130** - Constantine M. Dafermos,
*Hyperbolic conservation laws in continuum physics*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2000. MR**1763936**, DOI 10.1007/3-540-29089-3_{1}4 - P. Degond, T. Goudon, and F. Poupaud,
*Diffusion limit for nonhomogeneous and non-micro-reversible processes*, Indiana Univ. Math. J.**49**(2000), no. 3, 1175–1198. MR**1803225**, DOI 10.1512/iumj.2000.49.1936 - R. J. DiPerna,
*Convergence of approximate solutions to conservation laws*, Arch. Rational Mech. Anal.**82**(1983), no. 1, 27–70. MR**684413**, DOI 10.1007/BF00251724 - Ronald J. DiPerna,
*Compensated compactness and general systems of conservation laws*, Trans. Amer. Math. Soc.**292**(1985), no. 2, 383–420. MR**808729**, DOI 10.1090/S0002-9947-1985-0808729-4 - Donatella Donatelli and Pierangelo Marcati,
*Relaxation of semilinear hyperbolic systems with variable coefficients*, Ricerche Mat.**48**(1999), no. suppl., 295–310. Papers in memory of Ennio De Giorgi (Italian). MR**1765690** - Donatella Donatelli and Pierangelo Marcati,
*1-$\scr D$ relaxation from hyperbolic to parabolic systems with variable coefficients*, Rend. Istit. Mat. Univ. Trieste**31**(2000), no. suppl. 2, 63–85. Workshop on Blow-up and Global Existence of Solutions for Parabolic and Hyperbolic Problems (Trieste, 1999). MR**1800442** - S. D. Èĭdel′man,
*Parabolic systems*, North-Holland Publishing Co., Amsterdam-London; Wolters-Noordhoff Publishing, Groningen, 1969. Translated from the Russian by Scripta Technica, London. MR**0252806** - Lawrence C. Evans,
*Weak convergence methods for nonlinear partial differential equations*, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR**1034481**, DOI 10.1090/cbms/074 - Patrick Gérard,
*Microlocal defect measures*, Comm. Partial Differential Equations**16**(1991), no. 11, 1761–1794. MR**1135919**, DOI 10.1080/03605309108820822 - F. Golse, L. St. Raymond,
*The Navier-Stokes limit of the Boltzmann equation: convergence proof.*Preprint R01035, Laboratoire d’Analyse Numérique, Univ. Paris VI, 2001. - Lars Hörmander,
*The analysis of linear partial differential operators. I*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR**717035**, DOI 10.1007/978-3-642-96750-4 - Shi Jin and Hailiang Liu,
*Diffusion limit of a hyperbolic system with relaxation*, Methods Appl. Anal.**5**(1998), no. 3, 317–334. MR**1659135**, DOI 10.4310/MAA.1998.v5.n3.a6 - Shi Jin, Lorenzo Pareschi, and Giuseppe Toscani,
*Uniformly accurate diffusive relaxation schemes for multiscale transport equations*, SIAM J. Numer. Anal.**38**(2000), no. 3, 913–936. MR**1781209**, DOI 10.1137/S0036142998347978 - M. Junk, W.A. Yong,
*Rigorous Navier-Stokes Limit of the Lattice Boltzmann Equation*, Technical Report, IWR, Universität Heidelberg. - Markos A. Katsoulakis and Athanasios E. Tzavaras,
*Contractive relaxation systems and interacting particles for scalar conservation laws*, C. R. Acad. Sci. Paris Sér. I Math.**323**(1996), no. 8, 865–870 (English, with English and French summaries). MR**1414549** - Markos A. Katsoulakis and Athanasios E. Tzavaras,
*Contractive relaxation systems and the scalar multidimensional conservation law*, Comm. Partial Differential Equations**22**(1997), no. 1-2, 195–233. MR**1434144**, DOI 10.1080/03605309708821261 - Markos A. Katsoulakis and Athanasios E. Tzavaras,
*Multiscale analysis for interacting particles: relaxation systems and scalar conservation laws*, J. Statist. Phys.**96**(1999), no. 3-4, 715–763. MR**1716813**, DOI 10.1023/A:1004670308361 - Shuichi Kawashima,
*Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications*, Proc. Roy. Soc. Edinburgh Sect. A**106**(1987), no. 1-2, 169–194. MR**899951**, DOI 10.1017/S0308210500018308 - E. F. Keller, L. A. Segel,
*Initiation of slime mold aggregation viewed as instability*. J. Theor. Biol.**1970**, 399-415. - Heinz-Otto Kreiss,
*Initial boundary value problems for hyperbolic systems*, Comm. Pure Appl. Math.**23**(1970), 277–298. MR**437941**, DOI 10.1002/cpa.3160230304 - Heinz-Otto Kreiss and Jens Lorenz,
*Initial-boundary value problems and the Navier-Stokes equations*, Pure and Applied Mathematics, vol. 136, Academic Press, Inc., Boston, MA, 1989. MR**998379** - Thomas G. Kurtz,
*Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics*, Trans. Amer. Math. Soc.**186**(1973), 259–272 (1974). MR**336482**, DOI 10.1090/S0002-9947-1973-0336482-1 - Corrado Lattanzio and Pierangelo Marcati,
*The relaxation to the drift-diffusion system for the 3-D isentropic Euler-Poisson model for semiconductors*, Discrete Contin. Dynam. Systems**5**(1999), no. 2, 449–455. MR**1665756**, DOI 10.3934/dcds.1999.5.449 - Corrado Lattanzio and Roberto Natalini,
*Convergence of diffusive BGK approximations for nonlinear strongly parabolic systems*, Proc. Roy. Soc. Edinburgh Sect. A**132**(2002), no. 2, 341–358. MR**1899825**, DOI 10.1017/S0308210500001669 - C. Lattanzio, W.-A. Yong,
*Hyperbolic-Parabolic singular limits for first order nonlinear systems*. Comm. Partial Differential Equations (to appear). - Peter Lax,
*Shock waves and entropy*, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 603–634. MR**0393870** - R. Lefever, G. Nicolis,
*Chemical instabilities and sustained oscillations*. Journal of theoretical Biology,**30**, (1971), 267. - J.-L. Lions,
*Perturbations singulières dans les problèmes aux limites et en contrôle optimal*, Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin-New York, 1973 (French). MR**0600331** - Pierre Louis Lions and Giuseppe Toscani,
*Diffusive limit for finite velocity Boltzmann kinetic models*, Rev. Mat. Iberoamericana**13**(1997), no. 3, 473–513. MR**1617393**, DOI 10.4171/RMI/228 - Tai-Ping Liu,
*Hyperbolic conservation laws with relaxation*, Comm. Math. Phys.**108**(1987), no. 1, 153–175. MR**872145** - A. Majda,
*Compressible fluid flow and systems of conservation laws in several space variables*, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR**748308**, DOI 10.1007/978-1-4612-1116-7 - Andrew Majda and Stanley Osher,
*Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary*, Comm. Pure Appl. Math.**28**(1975), no. 5, 607–675. MR**410107**, DOI 10.1002/cpa.3160280504 - Pierangelo Marcati and Albert Milani,
*The one-dimensional Darcy’s law as the limit of a compressible Euler flow*, J. Differential Equations**84**(1990), no. 1, 129–147. MR**1042662**, DOI 10.1016/0022-0396(90)90130-H - Pierangelo Marcati, Albert J. Milani, and Paolo Secchi,
*Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system*, Manuscripta Math.**60**(1988), no. 1, 49–69. MR**920759**, DOI 10.1007/BF01168147 - Pierangelo Marcati and Roberto Natalini,
*Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation*, Arch. Rational Mech. Anal.**129**(1995), no. 2, 129–145. MR**1328473**, DOI 10.1007/BF00379918 - Pierangelo Marcati and Roberto Natalini,
*Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem*, Proc. Roy. Soc. Edinburgh Sect. A**125**(1995), no. 1, 115–131. MR**1318626**, DOI 10.1017/S030821050003078X - Pierangelo Marcati and Bruno Rubino,
*Hyperbolic to parabolic relaxation theory for quasilinear first order systems*, J. Differential Equations**162**(2000), no. 2, 359–399. MR**1751710**, DOI 10.1006/jdeq.1999.3676 - H. P. McKean,
*The central limit theorem for Carleman’s equation*, Israel J. Math.**21**(1975), no. 1, 54–92. MR**423553**, DOI 10.1007/BF02757134 - Guy Métivier,
*The block structure condition for symmetric hyperbolic systems*, Bull. London Math. Soc.**32**(2000), no. 6, 689–702. MR**1781581**, DOI 10.1112/S0024609300007517 - Sigeru Mizohata,
*The theory of partial differential equations*, Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara. MR**0599580** - François Murat,
*Compacité par compensation*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**5**(1978), no. 3, 489–507 (French). MR**506997** - G. Naldi, L. Pareschi,
*Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation*. Preprint. - G. Nicolis and I. Prigogine,
*Self-organization in nonequilibrium systems*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. From dissipative structures to order through fluctuations. MR**0522141** - James V. Ralston,
*Note on a paper of Kreiss*, Comm. Pure Appl. Math.**24**(1971), no. 6, 759–762. MR**606239**, DOI 10.1002/cpa.3160240603 - Bruno Rubino,
*Weak solutions to quasilinear wave equations of Klein-Gordon or sine-Gordon type and relaxation to reaction-diffusion equations*, NoDEA Nonlinear Differential Equations Appl.**4**(1997), no. 4, 439–457. MR**1485731**, DOI 10.1007/s000300050024 - Lawrence M. Graves,
*The Weierstrass condition for multiple integral variation problems*, Duke Math. J.**5**(1939), 656–660. MR**99** - Denis Serre,
*Relaxations semi-linéaire et cinétique des systèmes de lois de conservation*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**17**(2000), no. 2, 169–192 (French, with English and French summaries). MR**1753092**, DOI 10.1016/S0294-1449(99)00105-5 - L. Tartar,
*Compensated compactness and applications to partial differential equations*, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR**584398** - Luc Tartar,
*The compensated compactness method applied to systems of conservation laws*, Systems of nonlinear partial differential equations (Oxford, 1982) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 263–285. MR**725524** - Luc Tartar,
*$H$-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations*, Proc. Roy. Soc. Edinburgh Sect. A**115**(1990), no. 3-4, 193–230. MR**1069518**, DOI 10.1017/S0308210500020606 - Michael E. Taylor,
*Pseudodifferential operators*, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR**618463** - Michael E. Taylor,
*Pseudodifferential operators and nonlinear PDE*, Progress in Mathematics, vol. 100, Birkhäuser Boston, Inc., Boston, MA, 1991. MR**1121019**, DOI 10.1007/978-1-4612-0431-2 - Nelson Dunford,
*A mean ergodic theorem*, Duke Math. J.**5**(1939), 635–646. MR**98** - Wen-An Yong,
*Singular perturbations of first-order hyperbolic systems with stiff source terms*, J. Differential Equations**155**(1999), no. 1, 89–132. MR**1693210**, DOI 10.1006/jdeq.1998.3584

## Additional Information

**Donatella Donatelli**- Affiliation: Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, 67100 L’Aquila, Italy
- Email: donatell@univaq.it
**Pierangelo Marcati**- Affiliation: Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, 67100 L’Aquila, Italy
- Email: marcati@univaq.it
- Received by editor(s): July 15, 2002
- Received by editor(s) in revised form: March 26, 2003, and June 18, 2003
- Published electronically: January 6, 2004
- Additional Notes: This research was partially supported by EU financed network no. HPRN-CT-2002-00282 and by COFIN MIUR 2002 “Equazioni paraboliche e iperboliche nonlineari”
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**356**(2004), 2093-2121 - MSC (2000): Primary 35L40, 35K40; Secondary 58J45, 58J37
- DOI: https://doi.org/10.1090/S0002-9947-04-03526-3
- MathSciNet review: 2031055