## The Annals of Probability

### Degenerate parabolic stochastic partial differential equations: Quasilinear case

#### Abstract

In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an $L^{1}$-contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014–1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294–4336], the present result contains two new ingredients that provide simpler and more effective method of the proof: a generalized Itô formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.

#### Article information

Source
Ann. Probab., Volume 44, Number 3 (2016), 1916-1955.

Dates
Revised: November 2014
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.aop/1463410035

Digital Object Identifier
doi:10.1214/15-AOP1013

Mathematical Reviews number (MathSciNet)
MR3502597

Zentralblatt MATH identifier
1346.60094

#### Citation

Debussche, Arnaud; Hofmanová, Martina; Vovelle, Julien. Degenerate parabolic stochastic partial differential equations: Quasilinear case. Ann. Probab. 44 (2016), no. 3, 1916--1955. doi:10.1214/15-AOP1013. https://projecteuclid.org/euclid.aop/1463410035

#### References

• [1] Bauzet, C., Vallet, G. and Wittbold, P. A degenerate parabolic–hyperbolic Cauchy problem with a stochastic force. Preprint, hal-01003069.
• [2] Bauzet, C., Vallet, G. and Wittbold, P. (2012). The Cauchy problem for conservation laws with a multiplicative stochastic perturbation. J. Hyperbolic Differ. Equ. 9 661–709.
• [3] Brzeźniak, Z. and Ondreját, M. (2007). Strong solutions to stochastic wave equations with values in Riemannian manifolds. J. Funct. Anal. 253 449–481.
• [4] Carrillo, J. (1999). Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147 269–361.
• [5] Chen, G.-Q. and Perthame, B. (2003). Well-posedness for non-isotropic degenerate parabolic–hyperbolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 645–668.
• [6] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
• [7] Debussche, A. and Vovelle, J. (2010). Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259 1014–1042.
• [8] Denis, L., Matoussi, A. and Stoica, L. (2005). $L^{p}$ estimates for the uniform norm of solutions of quasilinear SPDE’s. Probab. Theory Related Fields 133 437–463.
• [9] Fellah, D. and Pardoux, É. (1992). Une formule d’Itô dans des espaces de Banach, et application. In Stochastic Analysis and Related Topics (Silivri, 1990). Progress in Probability 31 197–209. Birkhäuser, Boston, MA.
• [10] Feng, J. and Nualart, D. (2008). Stochastic scalar conservation laws. J. Funct. Anal. 255 313–373.
• [11] Gagneux, G. and Madaune-Tort, M. (1996). Analyse Mathématique de Modèles Non Linéaires de L’ingénierie Pétrolière. Mathématiques & Applications (Berlin) [Mathematics & Applications] 22. Springer, Berlin.
• [12] Hofmanová, M. (2013). Degenerate parabolic stochastic partial differential equations. Stochastic Process. Appl. 123 4294–4336.
• [13] Hofmanová, M. (2013). Strong solutions of semilinear stochastic partial differential equations. NoDEA Nonlinear Differential Equations Appl. 20 757–778.
• [14] Hofmanová, M. and Seidler, J. (2012). On weak solutions of stochastic differential equations. Stoch. Anal. Appl. 30 100–121.
• [15] Imbert, C. and Vovelle, J. (2004). A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications. SIAM J. Math. Anal. 36 214–232 (electronic).
• [16] Kim, J. U. (2003). On a stochastic scalar conservation law. Indiana Univ. Math. J. 52 227–256.
• [17] Kružkov, S. N. (1970). First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 228–255.
• [18] Ladyženskaja, O. A., Solonnikov, V. A. and Ural’ceva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23. Amer. Math. Soc., Providence, RI.
• [19] Lions, P. L., Perthame, B. and Souganidis, P. E. (2011/12). Stochastic averaging lemmas for kinetic equations. In Seminaire Equations Aux Derivee Partielles (Ecole Polytechnique). Available at arXiv:1204.0317.
• [20] Lions, P. L., Perthame, B. and Souganidis, P. E. (2013). Scalar conservation laws with rough (stochastic) fluxes. Stochastic Partial Differential Equations: Analysis and Computations 1 664–686.
• [21] Lions, P.-L., Perthame, B. and Tadmor, E. (1991). Formulation cinétique des lois de conservation scalaires multidimensionnelles. C. R. Acad. Sci. Paris Sér. I Math. 312 97–102.
• [22] Lions, P.-L., Perthame, B. and Tadmor, E. (1994). A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7 169–191.
• [23] Málek, J., Nečas, J., Rokyta, M. and R$\mathop{\mathrm{u}}\limits^{\circ}$žička, M. (1996). Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation 13. Chapman & Hall, London.
• [24] Ondreját, M. (2010). Stochastic nonlinear wave equations in local Sobolev spaces. Electron. J. Probab. 15 1041–1091.
• [25] Perthame, B. (1998). Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure. J. Math. Pures Appl. 77 1055–1064.
• [26] Perthame, B. (2002). Kinetic Formulation of Conservation Laws. Oxford Lecture Series in Mathematics and Its Applications 21. Oxford Univ. Press, Oxford.
• [27] Vallet, G. and Wittbold, P. (2009). On a stochastic first-order hyperbolic equation in a bounded domain. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 613–651.