Advertisement

Regularization and Well-Posedness by Noise for Ordinary and Partial Differential Equations

  • Benjamin GessEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

Abstract

We give a brief introduction and overview of the topic of regularization and well-posedness by noise for ordinary and partial differential equations. The article is an attempt to outline in a concise fashion different directions of research in this field that have attracted attention in recent years. We close the article with a look on more recent developments in the field of nonlinear SPDE, focusing on stochastic scalar conservation laws and porous media equations. The article is tailored at master/PhD level, trying to allow a smooth introduction to the subject and pointing at a large list of references to allow further in-depth study.

Keywords

Regularization and well-posedness by noise Stochastic scalar conservation laws Stochastic porous medium equation Stochastic hamilton-jacobi equation 

2000 Mathematics Subject Classification

60H15 35R60 35L65 

References

  1. 1.
    Barbu, V.: Nonlinear differential equations of monotone types in Banach spaces. Springer monographs in mathematics. Springer, New York (2010)Google Scholar
  2. 2.
    Liu, W., Röckner, M.: Stochastic partial differential equations: an introduction. Universitext. Springer, Cham (2015)Google Scholar
  3. 3.
    DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ambrosio, L.: Transport equation and Cauchy problem for \(BV\) vector fields. Invent. Math. 158(2), 227–260 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Crippa, G., De Lellis, C.: Estimates and regularity results for the diperna-lions flow. J. Reine Angew. Math. 616, 15–46 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ambrosio, L., Crippa, G.: Continuity equations and ODE flows with non-smooth velocity. Proc. R. Soc. Edinb. Sect. A 144(6), 1191–1244 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Flandoli, F.: Random perturbation of PDEs and fluid dynamic models. In: Lecture notes in mathematics, vol. 2015. Springer, Heidelberg (2011). Lectures from the 40th Probability Summer School held in Saint-Flour (2010)Google Scholar
  8. 8.
    Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. In: Graduate texts in mathematics, vol. 113, 2nd edn. Springer, New York (1991)Google Scholar
  9. 9.
    Revuz, D., Yor, M.: Fundamental principles of mathematical sciences. In: Continuous martingales and Brownian motion, vol 293 of Grundlehren der Mathematischen Wissenschaften, 3rd edn. Springer, Berlin (1999)Google Scholar
  10. 10.
    Veretennikov, A.J.: Strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. (N.S.), 111(153)(3):434–452, 480 (1980)Google Scholar
  11. 11.
    Menoukeu-Pamen, O., Meyer-Brandis, T., Nilssen, T., Proske, F., Zhang, T.: A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s. Math. Ann. 357(2), 761–799 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Meyer-Brandis, T., Proske, F.: Construction of strong solutions of SDE’s via Malliavin calculus. J. Funct. Anal. 258(11), 3922–3953 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Krylov, N.V., Röckner, M.: Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 131(2), 154–196 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fedrizzi, E., Flandoli, F.: Pathwise uniqueness and continuous dependence of SDEs with non-regular drift. Stochastics 83(3), 241–257 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Banos, D., Duedahl, S., Meyer-Brandis, T., Proske, F.: Construction of malliavin differentiable strong solutions of SDEs under an integrability condition on the drift without the yamada-watanabe principle, (2015). arXiv:1503.09019
  16. 16.
    Luo, D.: Quasi-invariance of the stochastic flow associated to Itô’s SDE with singular time-dependent drift. J. Theor. Probab. 28(4), 1743–1762 (2015)CrossRefGoogle Scholar
  17. 17.
    Flandoli, F., Gubinelli, M., Priola, E.: Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180(1), 1–53 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mohammed, S.-E.A., Nilssen, T.K., Proske, F.N.: Sobolev differentiable stochastic flows for SDEs with singular coefficients: applications to the transport equation. Ann. Probab. 43(3), 1535–1576 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bogachev, V.I., Krylov, N.V., Röckner, M., Shaposhnikov, S.V.: Fokker-Planck-Kolmogorov equations. In: Mathematical surveys and monographs, vol. 207. American Mathematical Society, Providence, RI (2015)Google Scholar
  20. 20.
    Figalli, A.: Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254(1), 109–153 (2008)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Le Bris, C., Lions, P.-L.: Existence and uniqueness of solutions to Fokker–Planck type equations with irregular coefficients. Commun. Partial Differ. Equ. 33(7–9), 1272–1317 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Trevisan, D.: Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. Electron. J. Probab. 21(22), 41 (2016)Google Scholar
  23. 23.
    Röckner, M., Zhang, X.: Weak uniqueness of Fokker-Planck equations with degenerate and bounded coefficients. C. R. Math. Acad. Sci. Paris 348(7–8), 435–438 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Haadem, S., Proske, F.: On the construction and Malliavin differentiability of solutions of Lévy noise driven SDE’s with singular coefficients. J. Funct. Anal. 266(8), 5321–5359 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Priola, E.: Pathwise uniqueness for singular SDEs driven by stable processes. Osaka J. Math. 49(2), 421–447 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Priola, E. Stochastic flow for SDEs with jumps and irregular drift term. In: Stochastic analysis, volume 105 of Banach Center Publications, pages 193–210. Polish Academy of Science Institute of Mathematics, Warsaw (2015)Google Scholar
  27. 27.
    Catellier, R., Gubinelli, M.: Averaging along irregular curves and regularisation of ODEs. Stoch. Process. Appl. 126(8), 2323–2366 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Davie, A.M. Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. IMRN, (24):Art. ID rnm124, 26 (2007)Google Scholar
  29. 29.
    Davie, A.M. Individual path uniqueness of solutions of stochastic differential equations. In: Stochastic analysis 2010, pp. 213–225. Springer, Heidelberg (2011)Google Scholar
  30. 30.
    Attanasio, S., Flandoli, F.: Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplication noise. Commun. Partial Differ. Equ. 36(8), 1455–1474 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Fedrizzi, E., Flandoli, F.: Noise prevents singularities in linear transport equations. J. Funct. Anal. 264(6), 1329–1354 (2013)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Flandoli, F., Gubinelli, M., Priola, E.: Does noise improve well-posedness of fluid dynamic equations? In: Stochastic partial differential equations and applications, vol. 25 of Quad. Mat., pages 139–155. Deptartment of Mathematics, Seconda University Napoli, Caserta, 2010Google Scholar
  33. 33.
    Flandoli, F., Gubinelli, M., Priola, E.: Remarks on the stochastic transport equation with Hölder drift. Rend. Semin. Mat. Univ. Politec. Torino 70(1), 53–73 (2012)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Maurelli, M. Thesis (2011)Google Scholar
  35. 35.
    Neves, W., Olivera, C.: Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition. NoDEA Nonlinear Differ. Equ. Appl. 22(5), 1247–1258 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Attanasio, S., Flandoli, F.: Zero-noise solutions of linear transport equations without uniqueness: an example. C. R. Math. Acad. Sci, Paris (2009)Google Scholar
  37. 37.
    Buckdahn, R., Ouknine, Y., Quincampoix, M.: On limiting values of stochastic differential equations with small noise intensity tending to zero. Bull. Sci. Math. 133(3), 229–237 (2009)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Delarue, F., Flandoli, F.: The transition point in the zero noise limit for a 1D Peano example. Discret. Contin. Dyn. Syst. 34(10), 4071–4083 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Pilipenko, A. Proske, F.: On a selection problem for small noise perturbation in multidimensional case, (2015). arXiv:1510.00966
  40. 40.
    Trevisan, D. Zero noise limits using local times. Electron. Commun. Probab. 18(31), 7 (2013)Google Scholar
  41. 41.
    Gyöngy, I., Pardoux, É.: On the regularization effect of space-time white noise on quasi-linear parabolic partial differential equations. Probab. Theory Relat. Fields 97(1–2), 211–229 (1993)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Butkovsky, O., Mytnik, L.: Regularization by noise and flows of solutions for a stochastic heat equation, 2016. arXiv:1610.02553
  43. 43.
    Nilssen, T.: Quasi-linear stochastic partial differential equations with irregular coefficients: Malliavin regularity of the solutions. Stoch. Partial Differ. Equ. Anal. Comput. 3(3), 339–359 (2015)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Flandoli, F., Romito. M.: Probabilistic analysis of singularities for the 3D Navier-Stokes equations. In: Proceedings of EQUADIFF, 10 (Prague, 2001), vol. 127, pages 211–218 (2002)Google Scholar
  45. 45.
    Flandoli, F., Romito, M.: Markov selections for the 3D stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 140(3–4), 407–458 (2008)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Math. 35(6), 771–831 (1982)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Delarue, F., Flandoli, F., Vincenzi, D.: Noise prevents collapse of Vlasov–Poisson point charges. Commun. Pure Appl. Math. 67(10), 1700–1736 (2014)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Majda, A.J., Bertozzi, A.L.: Vorticity and incompressible flow. In: Cambridge texts in applied mathematics, vol. 27. Cambridge University Press, Cambridge (2002)Google Scholar
  49. 49.
    Flandoli, F., Gubinelli, M., Priola, E.: Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations. Stoch. Process. Appl. 121(7), 1445–1463 (2011)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Fedrizzi, E., Flandoli, F., Priola, E., Vovelle, J.: Regularity of stochastic kinetic equations, (2016). arXiv:1606.01088
  51. 51.
    Cerrai, S.: Second order PDE’s in finite and infinite dimension: a probabilistic approach. Lecture notes in mathematics, vol. 1762. Springer, Berlin (2001)Google Scholar
  52. 52.
    Da Prato, G.: Kolmogorov equations for stochastic PDEs.In: Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel (2004)Google Scholar
  53. 53.
    Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. In: Encyclopedia of mathematics and its applications, vol. 44. Cambridge University Press, Cambridge (1992)Google Scholar
  54. 54.
    Da Prato, G., Flandoli, F.: Pathwise uniqueness for a class of SDE in Hilbert spaces and applications. J. Funct. Anal. 259(1), 243–267 (2010)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Da Prato, G., Flandoli, F., Priola, E., Röckner, M.: Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41(5), 3306–3344 (2013)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Da Prato, G., Flandoli, F., Priola, E., Röckner, M.: Strong uniqueness for stochastic evolution equations with unbounded measurable drift term. J. Theor. Probab. 28(4), 1571–1600 (2015)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Da Prato, G., Flandoli, F., Röckner, M.: Fokker-Planck equations for SPDE with non-trace-class noise. Commun. Math. Stat. 1(3), 281–304 (2013)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Da Prato, G., Flandoli, F., Röckner, M.: Uniqueness for continuity equations in Hilbert spaces with weakly differentiable drift. Stoch. Partial Differ. Equ. Anal. Comput. 2(2), 121–145 (2014)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Da Prato, G., Flandoli, F., Röckner, M., Veretennikov, A.Y.: Strong uniqueness for SDEs in Hilbert spaces with nonregular drift. Ann. Probab. 44(3), 1985–2023 (2016)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Dafermos, C.M.: Fundamental principles of mathematical sciences. In: Hyperbolic conservation laws in continuum physics, vol 325 of Grundlehren der Mathematischen Wissenschaften, 3rd edn. Springer, Berlin (2010)Google Scholar
  61. 61.
    Kružkov, S.N. First order quasilinear equations with several independent variables. Mat. Sb. (N.S.), 81(123), 228–255 (1970)Google Scholar
  62. 62.
    Lax, P.D.: Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537–566 (1957)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Crandall, M.G.: The semigroup approach to first order quasilinear equations in several space variables. Isr. J. Math. 12, 108–132 (1972)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Lions, P.-L., Perthame, B., Tadmor, E.: A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Am. Math. Soc. 7(1), 169–191 (1994)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Perthame, B.: Kinetic formulation of conservation laws. In: Oxford lecture series in mathematics and its applications, vol. 21. Oxford University Press, Oxford (2002)Google Scholar
  66. 66.
    Perthame, B., Souganidis, P.E.: Dissipative and entropy solutions to non-isotropic degenerate parabolic balance laws. Arch. Ration. Mech. Anal. 170(4), 359–370 (2003)MathSciNetCrossRefGoogle Scholar
  67. 67.
    Avellaneda, M., Weinan, E.: Statistical properties of shocks in Burgers turbulence. Commun. Math. Phys. 172(1), 13–38 (1995)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Burgers, J.: The non-linear diffusion equation: asymptotic solutions and statistical problems. Springer, Lecture series (1974)Google Scholar
  69. 69.
    Ryan, R.: Large-deviation analysis of Burgers turbulence with white-noise initial data. Commun. Pure Appl. Math. 51(1), 47–75 (1998)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Sinaĭ, Y.G.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys. 148(3), 601–621 (1992)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Nakazawa, H.: Stochastic Burgers’ equation in the inviscid limit. Adv. Appl. Math. 3(1), 18–42 (1982)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Weinan, E., Khanin,K., Mazel, A., Sinai, Y.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math. (2). 151(3), 877–960 (2000)Google Scholar
  73. 73.
    Debussche, A., Vovelle, J.: Invariant measure of scalar first-order conservation laws with stochastic forcing. Probab. Theory Relat. Fields 163(3–4), 575–611 (2015)MathSciNetCrossRefGoogle Scholar
  74. 74.
    Kim, J.U.: On a stochastic scalar conservation law. Indiana Univ. Math. J. 52(1), 227–256 (2003)MathSciNetCrossRefGoogle Scholar
  75. 75.
    Saussereau, B., Stoica, I.L.: Scalar conservation laws with fractional stochastic forcing: existence, uniqueness and invariant measure. Stoch. Process. Appl. 122(4), 1456–1486 (2012)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Vallet, G., Wittbold, P.: On a stochastic first-order hyperbolic equation in a bounded domain. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12(4), 613–651 (2009)MathSciNetCrossRefGoogle Scholar
  77. 77.
    Holden, H., Risebro, N.H.: Conservation laws with a random source. Appl. Math. Optim. 36(2), 229–241 (1997)MathSciNetCrossRefGoogle Scholar
  78. 78.
    Feng, J., Nualart, D.: Stochastic scalar conservation laws. J. Funct. Anal. 255(2), 313–373 (2008)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Debussche, A., Vovelle, J.: Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259(4), 1014–1042 (2010)MathSciNetCrossRefGoogle Scholar
  80. 80.
    Hofmanová, M.: Degenerate parabolic stochastic partial differential equations. Stoch. Process. Appl. 123(12), 4294–4336 (2013)MathSciNetCrossRefGoogle Scholar
  81. 81.
    Chen, G.-Q., Ding, Q., Karlsen, K.H.: On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal. 204(3), 707–743 (2012)MathSciNetCrossRefGoogle Scholar
  82. 82.
    Bauzet, C., Vallet, G., Wittbold, P.: The cauchy problem for conservation laws with a multiplicative stochastic perturbation. J. Hyperbolic Differ. Equ. 9(4), 661–709 (2013)MathSciNetCrossRefGoogle Scholar
  83. 83.
    Bauzet, C., Vallet, G., Wittbold, P.: The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation. J. Funct. Anal. 266(4), 2503–2545 (2014)MathSciNetCrossRefGoogle Scholar
  84. 84.
    Bauzet, C., Vallet, G., Wittbold, P.: A degenerate parabolic-hyperbolic Cauchy problem with a stochastic force. J. Hyperbolic Differ. Equ. 12(3), 501–533 (2015)MathSciNetCrossRefGoogle Scholar
  85. 85.
    Bauzet, C., Vallet, G., Wittbold, P., Zimmermann, A.: On a \(p(t, x)\)-Laplace evolution equation with a stochastic force. Stoch. Partial Differ. Equ. Anal. Comput. 1(3), 552–570 (2013)MathSciNetzbMATHGoogle Scholar
  86. 86.
    Kim, J.U.: On the Cauchy problem for the transport equation with random noise. J. Funct. Anal. 259(12), 3328–3359 (2010)MathSciNetCrossRefGoogle Scholar
  87. 87.
    Gess, B., Souganidis, P.E.: Scalar conservation laws with multiple rough fluxes. Commun. Math. Sci. 13(6), 1569–1597 (2015)MathSciNetCrossRefGoogle Scholar
  88. 88.
    Lions, P.-L., Perthame, B., Souganidis, P.E.: Scalar conservation laws with rough (stochastic) fluxes. Stoch. Partial Differ. Equ. Anal. Comput. 1(4), 664–686 (2013)MathSciNetzbMATHGoogle Scholar
  89. 89.
    Lions, P.-L., Perthame, B., Souganidis, P.E.: Stochastic averaging lemmas for kinetic equations. In: Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2011–2012, Sémin. Équ. Dériv. Partielles, pages Exp. No. XXVI, 17. École Polytech., Palaiseau (2013)Google Scholar
  90. 90.
    Lions, P.-L., Perthame, B., Souganidis, P.E.: Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case. Stoch. Partial Differ. Equ. Anal. Comput. 2(4), 517–538 (2014)MathSciNetzbMATHGoogle Scholar
  91. 91.
    Gess, B., Maurelli, M.: Well-posedness by noise for scalar conservation laws, (2017). arXiv:1701.05393
  92. 92.
    Andreianov, B., Karlsen, K.H., Risebro, N.H.: On vanishing viscosity approximation of conservation laws with discontinuous flux. Netw. Heterog. Media 5(3), 617–633 (2010)MathSciNetCrossRefGoogle Scholar
  93. 93.
    Andreianov, B., Karlsen, K.H., Risebro, N.H.: A theory of \(L^1\)-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal. 201(1), 27–86 (2011)MathSciNetCrossRefGoogle Scholar
  94. 94.
    Andreianov, B., Mitrović, D.: Entropy conditions for scalar conservation laws with discontinuous flux revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(6), 1307–1335 (2015)MathSciNetCrossRefGoogle Scholar
  95. 95.
    Crasta, G., De Cicco, V., De Philippis, G.: Kinetic formulation and uniqueness for scalar conservation laws with discontinuous flux. Commun. Partial Differ. Equ. 40(4), 694–726 (2015)MathSciNetCrossRefGoogle Scholar
  96. 96.
    Crasta, G., De Cicco, V., De Philippis, G., Ghiraldin, F.: Structure of solutions of multidimensional conservation laws with discontinuous flux and applications to uniqueness. Arch. Ration. Mech. Anal. 221(2), 961–985 (2016)MathSciNetCrossRefGoogle Scholar
  97. 97.
    Gassiat, P., Gess, B. Regularization by noise for stochastic Hamilton–Jacobi equations, (2016). arXiv:1609.07074
  98. 98.
    Gess, B., Souganidis, P.E.: Long-time behavior, invariant measures, and regularizing effects for stochastic scalar conservation laws. Commun. Pure Appl. Math. 70(8), 1562–1597 (2017)MathSciNetCrossRefGoogle Scholar
  99. 99.
    Gess, B., Souganidis, P.E.: Long-time behaviour, invariant measures and regularizing effects for stochastic scalar conservation laws - revised version, (2017) (preprint)Google Scholar
  100. 100.
    Golse, F., Perthame, B.: Optimal regularizing effect for scalar conservation laws. Rev. Mat. Iberoam. 29(4), 1477–1504 (2013)MathSciNetCrossRefGoogle Scholar
  101. 101.
    Jabin, P.-E., Perthame, B. Regularity in kinetic formulations via averaging lemmas. ESAIM Control Optim. Calc. Var. 8, 761–774 (2002) (electronic). A tribute to J. L. LionsGoogle Scholar
  102. 102.
    De Lellis, C., Westdickenberg, M.: On the optimality of velocity averaging lemmas. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(6), 1075–1085 (2003)MathSciNetCrossRefGoogle Scholar
  103. 103.
    Vázquez, J.L.: The porous medium equation: mathematical theory. Oxford mathematical monographs. The Clarendon Press Oxford University Press, Oxford (2007)Google Scholar
  104. 104.
    Gianazza, U., Schwarzacher, S.: Self-improving property of degenerate parabolic equations of porous medium-type, (2016). arXiv:1603.07241
  105. 105.
    Friz, P.K., Gassiat, P., Lions, P.-L., Souganidis, P.E.: Eikonal equations and pathwise solutions to fully non-linear SPDEs, (2016). arXiv:1602.04746
  106. 106.
    Lions, P.-L., Souganidis, P.E.: Stochastic viscosity solutions. (Book, in preparation)Google Scholar
  107. 107.
    Lions, P.-L.: Generalized solutions of Hamilton–Jacobi equations. In: Research notes in mathematics, vol. 69. Pitman (Advanced Publishing Program), Boston, Mass-London (1982)Google Scholar
  108. 108.
    Lasry, J.-M., Lions, P.-L: A remark on regularization in Hilbert spaces. Israel J. Math. 55(3) (1986)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Max-Planck Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations