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Gess, Benjamin; Hofmanová, Martina

Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE. (English) Zbl 1428.60090

Ann. Probab. 46, No. 5, 2495-2544 (2018).
Summary: We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full \(L^{1}\) setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an \(L^{1}\)-contraction property for the solutions, generalizing the results obtained in [A. Debussche et al., Ann. Probab. 44, No. 3, 1916–1955 (2016; Zbl 1346.60094)].

Cited in 31 Documents

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations

Keywords:

quasilinear degenerate parabolic stochastic partial differential equation; kinetic formulation; kinetic solution; velocity averaging lemmas; renormalized solutions

Citations:

Zbl 1346.60094
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Full Text: DOI arXiv Euclid

References:

[1] Ammar, K. and Wittbold, P. (2003). Existence of renormalized solutions of degenerate elliptic-parabolic problems. Proc. Roy. Soc. Edinburgh Sect. A133 477–496. · Zbl 1077.35103
[2] Bahouri, H., Chemin, J.-Y. and Danchin, R. (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin. · Zbl 1227.35004
[3] Barbu, V., Da Prato, G. and Röckner, M. (2008). Existence and uniqueness of nonnegative solutions to the stochastic porous media equation. Indiana Univ. Math. J.57 187–211. · Zbl 1137.76059 · doi:10.1512/iumj.2008.57.3284
[4] 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. · Zbl 1263.35222
[5] Bauzet, C., Vallet, G. and Wittbold, P. (2015). A degenerate parabolic-hyperbolic Cauchy problem with a stochastic force. J. Hyperbolic Differ. Equ.12 501–533. · Zbl 1327.35209 · doi:10.1142/S0219891615500150
[6] Berthelin, F. and Vovelle, J. (2010). A Bhatnagar–Gross–Krook approximation to scalar conservation laws with discontinuous flux. Proc. Roy. Soc. Edinburgh Sect. A140 953–972. · Zbl 1207.35217
[7] Bouchut, F. and Desvillettes, L. (1999). Averaging lemmas without time Fourier transform and application to discretized kinetic equations. Proc. Roy. Soc. Edinburgh Sect. A129 19–36. · Zbl 0933.35159 · doi:10.1017/S030821050002744X
[8] Carrillo, J. (1999). Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal.147 269–361. · Zbl 0935.35056 · doi:10.1007/s002050050152
[9] Carrillo, J. and Wittbold, P. (1999). Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems. J. Differential Equations156 93–121. · Zbl 0932.35129 · doi:10.1006/jdeq.1998.3597
[10] Chen, C. Q., Ding, Q. and Karlsen, K. H. (2012). On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal.204 707–743. · Zbl 1261.60062 · doi:10.1007/s00205-011-0489-9
[11] Chen, G.-Q. and Perthame, B. (2003). Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire20 645–668. · Zbl 1031.35077 · doi:10.1016/S0294-1449(02)00014-8
[12] Crandall, M. G. (1972). The semigroup approach to first order quasilinear equations in several space variables. Israel J. Math.12 108–132. · Zbl 0246.35018 · doi:10.1007/BF02764657
[13] Crandall, M. G. and Liggett, T. M. (1971). Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math.93 265–298. · Zbl 0226.47038 · doi:10.2307/2373376
[14] Da Prato, G., Röckner, M., Rozovskii, B. L. and Wang, F.-Y. (2006). Strong solutions of stochastic generalized porous media equations: Existence, uniqueness, and ergodicity. Comm. Partial Differential Equations31 277–291. · Zbl 1158.60356
[15] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia Math. Appl.44. Cambridge Univ. Press, Cambridge. · Zbl 0761.60052
[16] Debussche, A., Hofmanová, M. and Vovelle, J. (2016). Degenerate parabolic stochastic partial differential equations: Quasilinear case. Ann. Probab.44 1916–1955. DOI:10.1214/15-AOP1013. · Zbl 1346.60094
[17] Debussche, A. and Vovelle, J. (2015). Invariant measure of scalar first-order conservation laws with stochastic forcing. Probab. Theory Related Fields163 575–611. · Zbl 1331.60117 · doi:10.1007/s00440-014-0599-z
[18] Debussche, A. and Vovelle, J. Scalar conservation laws with stochastic forcing, revised version. Available at http://math.univ-lyon1.fr/vovelle/DebusscheVovelleRevised.pdf. · Zbl 1200.60050 · doi:10.1016/j.jfa.2010.02.016
[19] Debussche, A. and Vovelle, J. (2010). Scalar conservation laws with stochastic forcing. J. Funct. Anal.259 1014–1042. · Zbl 1200.60050 · doi:10.1016/j.jfa.2010.02.016
[20] Deya, A., Gubinelli, M., Hofmanová, M. and Tindel, S. General a priori estimates for rough PDEs with application to rough conservation laws. Available at arXiv:1604.00437. · Zbl 1255.60106
[21] DiPerna, R. J. and Lions, P. L. (1989). On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. of Math.130 321-366. · Zbl 0698.45010 · doi:10.2307/1971423
[22] DiPerna, R. J., Lions, P. L. and Meyer, Y. (1991). \(L^{p}\) regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire8 271–287. · Zbl 0763.35014
[23] Es-Sarhir, A. and von Renesse, M.-K. (2012). Ergodicity of stochastic curve shortening flow in the plane. SIAM J. Math. Anal.44 224–244. · Zbl 1251.47041 · doi:10.1137/100798235
[24] Feng, J. and Nualart, D. (2008). Stochastic scalar conservation laws. J. Funct. Anal.255 313–373. · Zbl 1154.60052 · doi:10.1016/j.jfa.2008.02.004
[25] Friz, P. K. and Gess, B. (2016). Stochastic scalar conservation laws driven by rough paths. Ann. Inst. H. Poincaré Anal. Non Linéaire33 933–963. · Zbl 1345.60055
[26] Gess, B. and Souganidis, P. E. (2016). Long-time behavior, invariant measures, and regularizing effects for stochastic scalar conservation laws. Comm. Pure Appl. Math.70 1562–1597. · Zbl 1370.60105 · doi:10.1002/cpa.21646
[27] Gess, B. and Souganidis, P. E. (2017). Stochastic non-isotropic degenerate parabolic-hyperbolic equations. Stochastic Process. Appl.127 2961–3004. · Zbl 1372.60090 · doi:10.1016/j.spa.2017.01.005
[28] Gess, B. and Souganidis, P. E. (2015). Scalar conservation laws with multiple rough fluxes. Commun. Math. Sci.13 1569–1597. · Zbl 1329.60210 · doi:10.4310/CMS.2015.v13.n6.a10
[29] Gess, B. and Tölle, J. M. (2014). Multi-valued, singular stochastic evolution inclusions. J. Math. Pures Appl.9 789–827. · Zbl 1295.47042 · doi:10.1016/j.matpur.2013.10.004
[30] Gess, B. and Tölle, J. M. (2016). Ergodicity and local limits for stochastic local and nonlocal \(p\)-Laplace equations. SIAM J. Math. Anal.48 4094–4125. · Zbl 1364.60082 · doi:10.1137/15M1049774
[31] Grafakos, L. (2008). Classical Fourier Analysis, 2nd ed. Graduate Texts in Mathematics249. Springer, New York. · Zbl 1220.42001
[32] Hofmanová, M. (2013). Degenerate parabolic stochastic partial differential equations. Stochastic Process. Appl.123 4294–4336. · Zbl 1291.60130
[33] Hofmanová, M. (2015). A Bhatnagar–Gross–Krook approximation to stochastic scalar conservation laws. Ann. Inst. Henri Poincaré Probab. Stat.51 1500–1528. · Zbl 1329.60214 · doi:10.1214/14-AIHP610
[34] Hofmanová, M. (2016). Scalar conservation laws with rough flux and stochastic forcing. Stoch. Partial Differ. Equ., Anal. Computat.4 635–690. · Zbl 1351.60080 · doi:10.1007/s40072-016-0072-3
[35] Holden, H. and Risebro, N. H. (1997). Conservation laws with a random source. Appl. Math. Optim.36 229–241. · Zbl 0885.35069 · doi:10.1007/BF02683344
[36] 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. · Zbl 1085.35099 · doi:10.1137/S003614100342468X
[37] Jabin, P. E. and Perthame, B. (2002). Regularity in kinetic formulations via averaging lemmas. ESAIM Control Optim. Calc. Var.8 761–774 (electronic). · Zbl 1065.35185
[38] Kim, J. U. (2003). On a stochastic scalar conservation law. Indiana Univ. Math. J.52 227–256. · Zbl 1037.60064 · doi:10.1512/iumj.2003.52.2310
[39] Komorowski, T., Peszat, S. and Szarek, T. (2010). On ergodicity of some Markov processes. Ann. Probab.38 1401–1443. · Zbl 1214.60035 · doi:10.1214/09-AOP513
[40] Kružkov, S. N. (1970). First order quasilinear equations with several independent variables. Mat. Sb.81 228–255. · Zbl 0202.11203
[41] Lions, P. L., Perthame, B. and Souganidis, P. E. (2013). Scalar conservation laws with rough (stochastic) fluxes. Stoch. Partial Differ. Equ., Anal. Computat.1 664–686. · Zbl 1333.60140 · doi:10.1007/s40072-013-0021-3
[42] Lions, P. L., Perthame, B. and Tadmor, E. (1991). Formulation cinétique des lois de conservation scalaires multidimensionnelles. C. R. Math. Acad. Sci. Paris 97–102. Série I. · Zbl 0729.49007
[43] 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. · Zbl 0820.35094 · doi:10.1090/S0894-0347-1994-1201239-3
[44] Lions, P.-L., Perthame, B. and Souganidis, P. E. (2014). Scalar conservation laws with rough (stochastic) fluxes: The spatially dependent case. Stoch. Partial Differ. Equ., Anal. Computat.2 517–538. · Zbl 1323.35233 · doi:10.1007/s40072-014-0038-2
[45] Perthame, B. (2002). Kinetic Formulation of Conservation Laws. Oxford Lecture Ser. Math. Appl.21. Oxford Univ. Press, Oxford. · Zbl 1030.35002
[46] Perthame, B. and Tadmor, E. (1991). A kinetic equation with kinetic entropy functions for scalar conservation laws. Comm. Math. Phys.136 501–517. · Zbl 0729.76070 · doi:10.1007/BF02099071
[47] Prévôt, C. and Röckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Math.1905. Springer, Berlin.
[48] Ren, J., Röckner, M. and Wang, F.-Y. (2007). Stochastic generalized porous media and fast diffusion equations. J. Differential Equations238 118–152. · Zbl 1129.60059
[49] Röckner, M. and Wang, F.-Y. (2008). Non-monotone stochastic generalized porous media equations. J. Differential Equations245 3898–3935. · Zbl 1151.76032 · doi:10.1016/j.jde.2008.03.003
[50] Saussereau, B. and Stoica, I. L. (2012). Scalar conservation laws with fractional stochastic forcing: Existence, uniqueness and invariant measure. Stochastic Process. Appl.122 1456–1486. · Zbl 1278.35158 · doi:10.1016/j.spa.2012.01.005
[51] Tadmor, E. and Tao, T. (2007). Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs. Comm. Pure Appl. Math.60 1488–1521. · Zbl 1131.35004 · doi:10.1002/cpa.20180
[52] Triebel, H. (1978). Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library18. North-Holland, Amsterdam. · Zbl 0387.46033
[53] 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. · Zbl 1194.60042 · doi:10.1142/S0219025709003872
[54] Vázquez, J.
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