On Some Recent Methods for Nonlinear Partial Differential Equations

  • Pierre-Louis Lions


We wish to present here some aspects of a few general methods that have been introduced recently in order to solve nonlinear partial differential equations and related problems in nonlinear analysis.


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  1. [1]
    H. Andréasson, A regularity property and strong L 1 convergence to equilibrium for the relativistic Boltzmann equation, preprint 21, Chalmers Univ., Göteborg, 1994.Google Scholar
  2. [2]
    L. Arkeryd and C. Cercignani, On the convergence of solutions of Enskog equations to solutions of the Boltzmann equation, Comm. Partial Differential Equations, 14 (1989), 1071–1090.MathSciNetCrossRefGoogle Scholar
  3. [3]
    J. Ball and R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal., 100 (1987), 13–52.MathSciNetCrossRefGoogle Scholar
  4. [4]
    C. Bardos, F. Golse, and D. Levermore, Fluid dynamics limits of kinetic equations, I, J. Statist. Phys., 63 (1991), 323–344; II, Comm. Pure Appl. Math., 46 (1993), 667–753MathSciNetCrossRefGoogle Scholar
  5. [5]
    M. Bézard, Régularité L Pprécisée des moyennes dans les equations de transport, preprint.Google Scholar
  6. [6]
    L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsberichte der Akademie der Wisssenschaften, Vienna, 66 (1972), 275–370. (Trans.: Further studies on the thermal equilibrium of gas molecules, in Kinetic Theory, vol. 2 (S. G. Brush, ed.), Pergamon, Oxford (1966), 88–174).zbMATHGoogle Scholar
  7. [7]
    L. Caffarelli, R. V. Kohn, and L. Nirenberg, On the regularity of the solutions of Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771–832.MathSciNetCrossRefGoogle Scholar
  8. [8]
    T. Carleman, Acta Math., 60 (1933), 91.MathSciNetCrossRefGoogle Scholar
  9. [9]
    T. Carleman, Problèmes mathématiques dans la théorie cinétique des gaz. Notes written by Carleson and Frostman, Uppsala, Almqvist and Wikselles, 1957.zbMATHGoogle Scholar
  10. [10]
    C. Cercignani, The Boltzmann Equation and its Applications. Springer Verlag, Berlin and New York, 1988.CrossRefGoogle Scholar
  11. [11]
    G. Q. Chen, The theory of compensated compactness and the system of isentropic gas dynamics, preprint.Google Scholar
  12. [12]
    R. Coifman, P. L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9), 72 (1993), 247–286.MathSciNetzbMATHGoogle Scholar
  13. [13]
    R. Coifman, R. Rochberg, and G. Weiss, Ann. of Math. (2), 103 (1976), 611–635.MathSciNetCrossRefGoogle Scholar
  14. [14]
    R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 579–645.MathSciNetCrossRefGoogle Scholar
  15. [15]
    M. G. Crandall and P. L. Lions, Hamilton-Jacobi equations in infinite dimensions, I, J. Funct. Anal., 63 (1985), 379–396; II, J. Funct. Anal., 65 (1985), 308–400; III, J. Funct. Anal., 68 (1986),214–247 ; IV, J. Funct. Anal., 90 (1990), 237–283; V, J. Funct. Anal., 97 (1991), 417–465; VI, in Evolution Equations, Control Theory and Biomathematics (Ph. Clement and G. Lumer, eds.), Lecture Notes in Pure and Appl. Math. 155, Dekker, New York, 1994: VII, to appear in J. Funct. Anal.CrossRefGoogle Scholar
  16. [16]
    L. Desvillettes, About the regularizing properties of the non cut-off Kac equation, preprint, 1994.Google Scholar
  17. [17]
    R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys., 91 (1983), 27–70.MathSciNetCrossRefGoogle Scholar
  18. [18]
    R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2), 130 (1989), 312–366.MathSciNetCrossRefGoogle Scholar
  19. [19]
    R. J. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 62 (1989), 729–757.MathSciNetCrossRefGoogle Scholar
  20. [20]
    R. J. DiPerna and P. L. Lions, Global solutions of Boltzmann’s equation and the entropy inequality, Arch. Rational Mech. Anal., 114 (1991), 47–55.MathSciNetCrossRefGoogle Scholar
  21. [21]
    R. J. DiPerna, P. L. Lions, and Y. Meyer, LP-regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 8 (1991), 271–287.CrossRefGoogle Scholar
  22. [22]
    M. J. Esteban and B. Perthame, On the modified Enskog equation with elastic or inelastic collisions; Models with spin, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 8 (1991), 289–398.MathSciNetCrossRefGoogle Scholar
  23. [23]
    C. Fefferman and E. Stein, xP spaces of several variables, Acta Math., 228 (1972), 137–193.CrossRefGoogle Scholar
  24. [24]
    P. Gerard, Microlocal defect measures, Comm. Partial Differential Equations, 16 (1991), 1761–1794.MathSciNetCrossRefGoogle Scholar
  25. [25]
    P. Gerard, Mesures semi-classiques et ondes de Bloch, in Séminaire EDP, 1990–1991, Ecole Polytechnique, Palaiseau, 1991.Google Scholar
  26. [26]
    F. Golse, P. L. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the solutions of a transport equation, J. Funct. Anal., 76 (1988), 110–125.MathSciNetCrossRefGoogle Scholar
  27. [27]
    F. Golse, B. Perthame, and R. Sentis, Un résultat pour les equations de transport et application au calcul de la limite de la valeur propre principale d’un opérateur de transport, C.R. Acad. Sci. Paris Série I, 301 (1985), 341–344.zbMATHGoogle Scholar
  28. [28]
    H. Grad, Principles of the kinetic theory of gases, Handbuch der Physik, 12, Springer Verlag, Berlin (1958), 205–294.MathSciNetGoogle Scholar
  29. [29]
    K. Harndache, Global existence for ‘weak solutions for the initial boundary value problems of Boltzmann equation, Arch Rational Mech Anal 119 (1992) 309–353MathSciNetCrossRefGoogle Scholar
  30. [30]
    F. Helein, Regularite des applications faiblement harmoniques entre une surface et une variété riemanienne, C. R. Acad. Sci. Paris Série I, 312 (1990), 591–596.zbMATHGoogle Scholar
  31. [31]
    D. Hilbert, Begründung der kinetischen Gastheorie, Math. Ann. 72 (1912), 562–577.MathSciNetCrossRefGoogle Scholar
  32. [32]
    D. Hoff, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with non smooth initial data, Proc. Roy. Soc. Edin-burgh Sect. A, 103 (1986), 301–315.CrossRefGoogle Scholar
  33. [33]
    D. Hoff, Global existence for 1D compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc., 303 (1987), 169–181.MathSciNetzbMATHGoogle Scholar
  34. [34]
    D. Hoff, Global well-posedness of the Cauchy problem for non-isentropic gas dynam-ics with discontinuous initial data, J. Differential Equations, 95 (1992), 33–73.MathSciNetCrossRefGoogle Scholar
  35. [35]
    R. D. James and D. Kinderlehrer, Theory of diffusionless phase transformations, Lecture Notes in Phys. 344, (M. Rascle, D. Serre, and M. Slemod, eds.), Springer Verlag, Berlin and New York, 1989.Google Scholar
  36. [36]
    A. V. Kazhikov, Cauchy problem for viscous gas equations, Sibirsk. Mat. Zh., 23 (1982), 60–64.MathSciNetGoogle Scholar
  37. [37]
    A. V. Kazhikov and V. V. Shelukhin, Unique global solution with respect to time of the initial boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273–282.MathSciNetCrossRefGoogle Scholar
  38. [38]
    P. D. Lax, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math., 10 (1957). 537–566.MathSciNetCrossRefGoogle Scholar
  39. [39]
    P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations J. Math. Phys. 5 (1964) 611–613.MathSciNetCrossRefGoogle Scholar
  40. [40]
    P. D. Lax, Shock waves and entropy, in Contributions to Nonlinear Functional Analysis (Zarantonello, ed.), Academic Press, New York, (1973), 603–634.Google Scholar
  41. [41]
    P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves. CBMS-NSF Regional Conferences Series in Applied Mathematics, 11. 1973.Google Scholar
  42. [42]
    J. Leray, Etude de diverses equations intégrales nonlinéaires et de quelques problemes aue pose l’hydrodynamique, J. Math. Pures Appl. (9), 12 (1933). 1–82.zbMATHGoogle Scholar
  43. [43]
    J. Leray, Essai sur les mouvernents plans d’un liquide visqueux que limitent des varois, J. Math. Pures Appl. (9). 13 (1934). 331–418.zbMATHGoogle Scholar
  44. [44]
    J. Leray, Essai sur le mouvernent d’un liquide visqueux emplissant l’espace, Acta Math., 63 (1934), 193–248.MathSciNetCrossRefGoogle Scholar
  45. [45]
    P. L. Lions, Existence globale de solutions pour les equations de Navier-Stokes compressibles isentropiques, C. R. Acad. Sci. Paris Série I, 316 (1993), 1335–1340. Compacité des solutions des equations de Navier-Stokes compressibles isentropiques, C. R. Acad. Sci. Paris Serie I, 317 (1993), 115–120. Limites incompressibles et acoustique pour des fiuides visqueux compressibles isentropiques, C. R. Acad. Sci. Paris Serie I, 317 (1993), 1197–1202.zbMATHGoogle Scholar
  46. [46]
    P. L. Lions, Conditions at infinity for Boltzmann’s equation, Comm. Partial Differential Equations, 19 (1994), 335–367.MathSciNetCrossRefGoogle Scholar
  47. [47]
    P. L. Lions, On Boltzmann and Landau equations, Phil. Trans. Roy. Soc. London Ser. A, 346 (1994), 191–204.MathSciNetCrossRefGoogle Scholar
  48. [48]
    P. L. Lions, Compactness in Boltzmann’s equation via Fourier integral operators and applications. Parts I–III, to appear in J. Math. Kyoto Univ., 1994.Google Scholar
  49. [49]
    P. L. Lions, Mathematical Topics in Fluid Mechanics., to appear in Oxford Univ. Press.Google Scholar
  50. [50]
    P. L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553–618.MathSciNetCrossRefGoogle Scholar
  51. [51]
    P. L. Lions, B. Perthame, and P. E. Souganidis, Existence and compactness of entropy solutions for the one-dimensional isentropic gas dynamics systems, to appear in Comm. Pure Appl. Math.Google Scholar
  52. [52]
    P. L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc., 7 (1994), 169–191.MathSciNetCrossRefGoogle Scholar
  53. [53]
    P. L. Lions, B. Perthame, and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems, to appear in Comm. Math. Phys.Google Scholar
  54. [54]
    A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer, Berlin and New York, 1984.CrossRefGoogle Scholar
  55. [55]
    A. Majda, Mathematical fluid dynamics: The interaction of nonlinear analysis and modern applied mathematics, in Proceedings of the AMS Centennial Symposium, August 8–12, 1988.Google Scholar
  56. [56]
    A. Majda, The interaction of Nonlinear Analysis and Modern Applied Mathematics, in Proc. Internat. Congress Math., Kyoto, 1990, vol. I, Springer, Berlin and New York. 1991Google Scholar
  57. [57]
    J. C. Maxwell, Scientific papers. Vol. 2, Cambridge Univ. Press, Cambridge, 1880 (Reprinted by Dover Publications. New York. 1965).zbMATHGoogle Scholar
  58. [58]
    J. C. Maxwell, On the dynamical theory of gases, Phil. Trans. Roy. Soc. London Ser. A, 157 (1886), 49–88.CrossRefGoogle Scholar
  59. [59]
    S. Müller, A surprising higher integrability property of mappings with positive determinant Proc Amer Math Soc., 21 (1989) 245–248MathSciNetzbMATHGoogle Scholar
  60. [60]
    F. Murat, Compacite par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 5 (1978), 489–507; II, in Proceedings of the International Meeting on Recent Methods on Nonlinear Analysis (E. De Giorgi, E. Magenes, and U. Mosco, eds.), Pitagora, Bologna, 1979; III, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 8 (1981), 69–102.MathSciNetzbMATHGoogle Scholar
  61. [61]
    T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible equation, Comm. Math. Phys., 61 (1978), 119–148.MathSciNetCrossRefGoogle Scholar
  62. [62]
    D. H. Phong and E. Stein, Hilbert integrals, singular integrals and Radon transforms, Ann. of Math. (2).Google Scholar
  63. [63]
    D. Serre, Solutions faibles globales des equations de Navier-Stokes pour un fluide compressible, C. R. Acad. Sci. Paris Série I. 303 (1986). 629–642.zbMATHGoogle Scholar
  64. [64]
    D. Serre, Sur l’equation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur, C. R. Acad. Sci. Paris Série I, 303 (1986), 703–706.zbMATHGoogle Scholar
  65. [65]
    D. Serre, Variations de grande amplitude pour la densité d’un fluide visqueux compressible, Phys. D, 48 (1991), 113–128.MathSciNetCrossRefGoogle Scholar
  66. [66]
    E. Stein, Oscillatory integrals in Fourier analysis, in Beijing Lectures in Harmonic Analysis (E. Stein, ed.), Princeton Univ. Press, Princeton, NJ (1986), 307–355.Google Scholar
  67. [67]
    E. Stein and G. Weiss, On the theory of HP spaces, Acta Math., 103 (1960), 25–62.MathSciNetCrossRefGoogle Scholar
  68. [68]
    L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. 4 (R. J. Knops, ed.), Research Notes in Math., Pitman, London, 1979.Google Scholar
  69. [69]
    L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equations (J. M. Ball, ed.), NATO ASI Series C III, Reidel, New York, 1983.zbMATHGoogle Scholar
  70. [70]
    L. Tartar, H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 193–230.MathSciNetCrossRefGoogle Scholar
  71. [71]
    S. Ukai and K. Asano, The Euler limit and initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J., 12 (1983), 311–332.MathSciNetCrossRefGoogle Scholar
  72. [72]
    B. Wennberg, Regularity estimates for the Boltzmann equation, preprint 2, 1994, Chalmers Univ., Goteborg.zbMATHGoogle Scholar

Copyright information

© Birkhäser Verlag, Basel, Switzerland 1995

Authors and Affiliations

  • Pierre-Louis Lions
    • 1
  1. 1.CEREMADE, Université Paris-DauphineParis Cedex 16France

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