Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE

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.

[1]  J. Carrillo,et al.  Uniqueness of Renormalized Solutions of Degenerate Elliptic-Parabolic Problems , 1999 .

[2]  Guy Vallet,et al.  ON A STOCHASTIC FIRST-ORDER HYPERBOLIC EQUATION IN A BOUNDED DOMAIN , 2009 .

[3]  Michael G. Crandall,et al.  The semigroup approach to first order quasilinear equations in several space variables , 1972 .

[4]  Benjamin Gess,et al.  Ergodicity and Local Limits for Stochastic Local and Nonlocal p-Laplace Equations , 2015, SIAM J. Math. Anal..

[5]  P. Wittbold,et al.  Existence of renormalized solutions of degenerate elliptic-parabolic problems , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[6]  Arnaud Debussche,et al.  Degenerate parabolic stochastic partial differential equations: Quasilinear case , 2013, 1309.5817.

[7]  L. Grafakos.  Classical Fourier Analysis , 2010 .

[8]  J. Vázquez.  The Porous Medium Equation , 2006 .

[9]  Huijiang Zhao.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES WITH SINGULAR INITIAL DATA Lp(p , 1996 .

[10]  B. Saussereau,et al.  Scalar conservation laws with fractional stochastic forcing: Existence, uniqueness and invariant measure , 2012 .

[11]  Benoît Perthame,et al.  Kinetic formulation of conservation laws , 2002 .

[12]  Abdelhadi Es-Sarhir,et al.  Ergodicity of Stochastic Curve Shortening Flow in the Plane , 2010, SIAM J. Math. Anal..

[13]  Tomasz Szarek,et al.  On ergodicity of some Markov processes , 2008, 0810.4609.

[14]  F. Bouchut,et al.  Averaging lemmas without time Fourier transform and application to discretized kinetic equations , 1999, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[15]  Gui-Qiang G. Chen,et al.  On Nonlinear Stochastic Balance Laws , 2011, 1111.5217.

[16]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[17]  Pierre-Emmanuel Jabin,et al.  Regularity in kinetic formulations via averaging lemmas , 2002 .

[18]  A. Debussche,et al.  Scalar conservation laws with stochastic forcing , 2010, 1001.5415.

[19]  Formulation cinétique des lois de conservation scalaires mulditimensionnelles , 1991 .

[20]  P. Lions,et al.  On the Cauchy problem for Boltzmann equations: global existence and weak stability , 1989 .

[21]  Michael G. Crandall,et al.  GENERATION OF SEMI-GROUPS OF NONLINEAR TRANSFORMATIONS ON GENERAL BANACH SPACES, , 1971 .

[22]  Feng-Yu Wang,et al.  Stochastic generalized porous media and fast diffusion equations , 2006, math/0602369.

[23]  B. Perthame,et al.  A kinetic equation with kinetic entropy functions for scalar conservation laws , 1991 .

[24]  Michael Röckner,et al.  Strong Solutions of Stochastic Generalized Porous Media Equations: Existence, Uniqueness, and Ergodicity , 2005, math/0512259.

[25]  Panagiotis E. Souganidis,et al.  Scalar conservation laws with multiple rough fluxes , 2014, 1406.2978.

[26]  J. Vovelle,et al.  A Kinetic Formulation for Multidimensional Scalar Conservation Laws with Boundary Conditions and Applications , 2004, SIAM J. Math. Anal..

[27]  M. Röckner,et al.  A Concise Course on Stochastic Partial Differential Equations , 2007 .

[28]  Panagiotis E. Souganidis,et al.  Scalar conservation laws with rough (stochastic) fluxes , 2013, Stochastic Partial Differential Equations: Analysis and Computations.

[29]  Carl Graham.  Long‐time behavior , 2014 .

[30]  Pierre-Louis Lions,et al.  Lp regularity of velocity averages , 1991 .

[31]  D. Nualart,et al.  Stochastic scalar conservation laws , 2008 .

[32]  J. Carrillo.  Entropy Solutions for Nonlinear Degenerate Problems , 1999 .

[33]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[34]  M. Gubinelli,et al.  A priori estimates for rough PDEs with application to rough conservation laws , 2016, Journal of Functional Analysis.

[35]  Panagiotis E. Souganidis,et al.  Stochastic non-isotropic degenerate parabolic–hyperbolic equations , 2016, 1611.01303.

[36]  Michael Röckner,et al.  Non-monotone stochastic generalized porous media equations☆ , 2008 .

[37]  M. Hofmanová.  Degenerate parabolic stochastic partial differential equations , 2013 .

[38]  Giuseppe Da Prato,et al.  Existence and uniqueness of nonnegative solutions to the stochastic porous media equation , 2007 .

[39]  Guy Vallet,et al.  THE CAUCHY PROBLEM FOR CONSERVATION LAWS WITH A MULTIPLICATIVE STOCHASTIC PERTURBATION , 2012 .

[40]  H. Holden,et al.  Conservation laws with a random source , 1997 .

[41]  Florent Berthelin,et al.  A BGK approximation to scalar conservation laws with discontinuous flux , 2010, 1003.4155.

[42]  Arnaud Debussche,et al.  Invariant measure of scalar first-order conservation laws with stochastic forcing , 2013, 1310.3779.

[43]  P. Friz,et al.  Stochastic scalar conservation laws driven by rough paths , 2014, 1403.6785.

[44]  P. Souganidis,et al.  Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws , 2014, 1411.3939.

[45]  Eitan Tadmor,et al.  Velocity averaging, kinetic formulations, and regularizing effects in quasi‐linear PDEs , 2007 .

[46]  R. Danchin,et al.  Fourier Analysis and Nonlinear Partial Differential Equations , 2011 .

[47]  Martina Hofmanova,et al.  A Bhatnagar-Gross-Krook Approximation to Stochastic Scalar Conservation Laws , 2013, 1305.6450.

[48]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[49]  R. Schwarzenberger.  ORDINARY DIFFERENTIAL EQUATIONS , 1982 .

[50]  Gui-Qiang G. Chen,et al.  Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations , 2003 .

[51]  Guy Vallet,et al.  A degenerate parabolic–hyperbolic Cauchy problem with a stochastic force , 2015 .

[52]  Benjamin Gess,et al.  Multi-valued, singular stochastic evolution inclusions , 2011, 1112.5672.

[53]  M. Hofmanová.  Scalar conservation laws with rough flux and stochastic forcing , 2015, 1503.03631.

[54]  Panagiotis E. Souganidis,et al.  Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case , 2014, Stochastic Partial Differential Equations: Analysis and Computations.

[55]  J. U. Kim,et al.  On a stochastic scalar conservation law , 2003 .

[56]  B. Perthame,et al.  A kinetic formulation of multidimensional scalar conservation laws and related equations , 1994 .