A Boundary Value Problem for a Class of Anisotropic Stochastic Degenerate Parabolic-Hyperbolic Equations

We establish the well-posedness of an initial-boundary value problem of mixed type for a stochastic nonlinear parabolic-hyperbolic equation on a space domain O = O′×O′′ where a Neumann boundary condition is imposed on ∂O′ × O′′, the hyperbolic boundary, and a Dirichlet condition is imposed on O′ × ∂O′′, the parabolic boundary. Among other points to be highlighted in our analysis of this problem we mention the new strong trace theorem for the special class of stochastic nonlinear parabolic-hyperbolic equations studied here, which is decisive for the uniqueness of the kinetic solution, and the new averaging lemma for the referred class of equations which is a vital part of the proof of the strong trace property. We also provide a detailed analysis of the approximate nondegenerate problems, which is also made here for the first time, as far as the authors know, whose solutions we prove to converge to the solution of our initial-boundary value problem.

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

[2]  Katsushi Ikeuchi,et al.  Stochastic Differential Equations in Infinite Dimensions , 2014, Computer Vision, A Reference Guide.

[3]  I. Gyöngy,et al.  Existence of strong solutions for Itô's stochastic equations via approximations , 1996 .

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

[5]  Kenneth Hvistendahl Karlsen,et al.  On stochastic conservation laws and Malliavin calculus , 2015, 1507.05518.

[6]  Hermano Frid,et al.  On the theory of divergence-measure fields and its applications , 2001 .

[7]  L. Tubaro,et al.  An estimate of Burkholder type for stochastic processes defined by the stochastic integral , 1984 .

[8]  Leszek Gawarecki,et al.  Stochastic Differential Equations in Infinite Dimensions , 2011 .

[9]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

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

[11]  On stochastic partial differential equations with polynomial nonlinearities , 1999 .

[12]  M. Beiglböck,et al.  A short proof of the Doob–Meyer theorem , 2010, Stochastic processes and their applications.

[13]  Divergence-Measure Fields on Domains with Lipschitz Boundary , 2014 .

[14]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[15]  Hermano Frid,et al.  Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics , 2003 .

[16]  The strong trace property and the Neumann problem for stochastic conservation laws , 2021, Stochastics and Partial Differential Equations: Analysis and Computations.

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

[18]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[19]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[20]  D. Lieberman,et al.  Fourier analysis , 2004, Journal of cataract and refractive surgery.

[21]  Jöran Bergh,et al.  Interpolation Spaces: An Introduction , 2011 .

[22]  P. Kotelenez.  A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations , 1984 .

[23]  R. Cooke.  Real and Complex Analysis , 2011 .

[24]  Jessika Eichel,et al.  Partial Differential Equations Second Edition , 2016 .

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

[26]  Gui-Qiang G. Chen,et al.  Large-time behavior of periodic entropy solutions to anisotropic degenerate parabolic-hyperbolic equations , 2008, 0810.2862.

[27]  Martin Ondreját,et al.  Uniqueness for stochastic evolution equations in Banach spaces , 2004 .

[28]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[29]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[30]  Xiongzhi Chen.  Brownian Motion and Stochastic Calculus , 2008 .

[31]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

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

[33]  Hermano Frid,et al.  Divergence‐Measure Fields and Hyperbolic Conservation Laws , 1999 .

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

[35]  Friedrich Sauvigny,et al.  Linear Operators in Hilbert Spaces , 2012 .

[36]  M. Šilhavý.  The Divergence Theorem for Divergence Measure Vectorfields on Sets with Fractal Boundaries , 2009 .

[37]  E. Yu. Panov,et al.  EXISTENCE OF STRONG TRACES FOR QUASI-SOLUTIONS OF MULTIDIMENSIONAL CONSERVATION LAWS , 2007 .

[38]  Corrado Mascia,et al.  Nonhomogeneous Dirichlet Problems for Degenerate Parabolic-Hyperbolic Equations , 2002 .

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

[40]  P. Kotelenez.  A submartingale type inequality with applicatinos to stochastic evolution equations , 1982 .

[41]  B. Davis,et al.  Integral Inequalities for Convex Functions of Operators on Martingales , 2011 .

[42]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[43]  Alexis Vasseur,et al.  Strong Traces for Solutions of Multidimensional Scalar Conservation Laws , 2001 .

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

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

[46]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

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

[48]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[49]  R. Kadison,et al.  Fundamentals of the Theory of Operator Algebras , 1983 .

[50]  장윤희,et al.  Y. , 2003, Industrial and Labor Relations Terms.

[51]  J. Norris.  Appendix: probability and measure , 1997 .

[52]  Doctoral Thesis,et al.  Instituto de Matematica Pura e Aplicada , 2009 .

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

[54]  A. Davie.  Uniqueness of solutions of stochastic differential equations , 2007, 0709.4147.