April  2020, 19(4): 2289-2331. doi: 10.3934/cpaa.2020100

The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise

1. 

Department of Mathematics and The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA

2. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA

Dedicated to Professor Tomás Caraballo on the occasion of his sixtieth birthday

Received  June 2019 Revised  October 2019 Published  January 2020

In this work, we investigate the existence of positive (martingale and pathwise) solutions of stochastic partial differential equations (SPDEs) driven by a Lévy noise. The proof relies on the use of truncation, following the Stampacchia approach to maximum principle. Among the applications, the positivity and boundedness results for the solutions of some biological systems and reaction diffusion equations are provided under suitable hypotheses, as well as some comparison theorems. This article improves the results of [15] where the authors only considered the case of the Wiener noise; even in this case we improve on [15] because the coefficients of the principal differential operator are now allowed to depend upon $ t $.

Citation: Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100
References:
[1]

D. Aldous, Stopping times and tightness, Ann. Probability, 6 (1978), 335-340.  doi: 10.1214/aop/1176995579.  Google Scholar

[2]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.  Google Scholar

[3]

H. BessaihE. Hausenblas and P. A. Razafimandimby, Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1661-1697.  doi: 10.1007/s00030-015-0339-9.  Google Scholar

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P. Billingsley, Convergence of Probability Measures, $2^{nd}$ edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

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Z. BrzeźniakHa usenblas and J. Zhu, 2D stochastic Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139.  doi: 10.1016/j.na.2012.10.011.  Google Scholar

[6]

Z. BrzeźniakE. Hausenblas and P. A. Razafimandimby, Stochastic reaction-diffusion equations driven by jump processes, Potential Anal., 49 (2018), 131-201.  doi: 10.1007/s11118-017-9651-9.  Google Scholar

[7]

Z. Brzeniak, W. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Analysis: Real World Applications, 17 (2014), 283–310. doi: 10.1016/j.nonrwa.2013.12.005.  Google Scholar

[8]

N. Chen and A. J. Majda, Simple dynamical models capturing the key features of the Central Pacific El Niño, P. Natl. Acad. Sci. USA, 113 (2016), 11732-11737.   Google Scholar

[9]

Q. ChenC. Miao and Z. Zhang, On the well-posedness for the viscous shallow water equations, SIAM J. Math. Anal., 40 (2008), 443-474.  doi: 10.1137/060660552.  Google Scholar

[10]

J. Cyr, S. Tang and R. Temam, A comparison of two settings for stochastic integration with respect to Lévy processes in infinite dimensions, in Trends in Applications of Mathematics to Mechanics (eds. E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin), Springer INdAM Series, (2018), 289–373.  Google Scholar

[11]

Justin Cyr, Phuong Nguyen, Sisi Tang and Roger Temam, Review of local and global existence results for stochastic PDEs with Lévy noise, to appear. Google Scholar

[12]

Justin Cyr, Phuong Nguyen, Sisi Tang and Roger Temam, Stochastic one layer shallow equation with Lévy noise, Discrete and Continuous Dynamical Systems B, Special issue in honor of Peter Kloeden, 24 (2019), 3765–3818.  Google Scholar

[13]

Justin Cyr, Phuong Nguyen, Sisi Tang and Roger Temam, The Euler equations of an inviscid incompressible fluid driven a Lévy noise, Nonlinear Analysis: Real World Applications, 44 (2018), 173–222. doi: 10.1016/j.nonrwa.2018.04.002.  Google Scholar

[14]

Mickaël D. Chekroun and Lionel J. Roques, Models of population dynamics under the influence of external perturbations: mathematical results, Comptes Rendus Mathematique, 343 (2006), 307–310. doi: 10.1016/j.crma.2006.07.012.  Google Scholar

[15]

Mickaël D. Chekroun, Eunhee Park and Roger Temam, The Stampacchia maximum principle for stochastic partial differential equations and applications, J. Differential Equations, 260 (2016), 2926–2972. doi: 10.1016/j.jde.2015.10.022.  Google Scholar

[16]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, $2^{nd}$ edition, volume 152 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781107295513.  Google Scholar

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[19]

A. DebusscheN. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.  Google Scholar

[20]

A. DebusscheN. Glatt-HoltzR. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118.  doi: 10.1088/0951-7715/25/7/2093.  Google Scholar

[21]

B. Di MartinoF. J. Chatelon and P. Orenga, The nonlinear Galerkin method applied to shallow water equations, Math. Models Methods Appl. Sci., 9 (1999), 825-854.  doi: 10.1142/S0218202599000397.  Google Scholar

[22]

Z. Dong and Y. Xie, Global solutions of stochastic 2D Navier-Stokes equations with Lévy noise, Sci. China Ser. A, 52 (2009), 1497-1524.  doi: 10.1007/s11425-009-0124-5.  Google Scholar

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[24]

B. P. W. Fernando and S. S. Sritharan, Nonlinear filtering of stochastic Navier-Stokes equation with Itô-Lévy noise, Stoch. Anal. Appl., 31 (2013), 381-426.  doi: 10.1080/07362994.2013.759482.  Google Scholar

[25]

F. Flandoli and D. Gątarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[26]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.   Google Scholar

[27]

I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158.  doi: 10.1007/BF01203833.  Google Scholar

[28]

D. J. Higham and P. E. Kloeden, Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math., 205 (2007), 949-956.  doi: 10.1016/j.cam.2006.03.039.  Google Scholar

[29]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, $2^{nd}$ edition, volume 24 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.  Google Scholar

[30]

T. G. Kurtz, The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities, Electron. J. Probab., 12 (2007), 951-965.  doi: 10.1214/EJP.v12-431.  Google Scholar

[31]

J. LinkP. Nguyen and R. Temam, Local martingale solutions to the stochastic one layer shallow water equations, J. Math. Anal. Appl., 448 (2017), 93-139.  doi: 10.1016/j.jmaa.2016.10.036.  Google Scholar

[32]

M. Métivier, Semimartingales: A Course on Stochastic Processes, volume 2 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin-New York, 1982.  Google Scholar

[33]

M. Métivier, Stochastic Partial Differential Equations in Infinite-dimensional Spaces, Scuola Normale Superiore di Pisa. Quaderni. [Publications of the Scuola Normale Superiore of Pisa]. Scuola Normale Superiore, Pisa, 1988.  Google Scholar

[34]

E. Motyl, Stochastic Navier-Stokes equations driven by Lévy noise in unbounded 3D domains, Potential Anal., 38 (2013), 863-912.  doi: 10.1007/s11118-012-9300-2.  Google Scholar

[35]

E. Motyl, Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains–abstract framework and applications, Stochastic Process. Appl., 124 (2014), 2052-2097.  doi: 10.1016/j.spa.2014.01.009.  Google Scholar

[36]

P. Orenga, Un théorème d'existence de solutions d'un problème de shallow water, Arch. Rational Mech. Anal., 130 (1995), 183-204.  doi: 10.1007/BF00375155.  Google Scholar

[37]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, volume 113 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.  Google Scholar

[38]

M. Röckner and T. Zhang, Stochastic evolution equations of jump type: existence, uniqueness and large deviation principles, Potential Anal., 26 (2007), 255-279.  doi: 10.1007/s11118-006-9035-z.  Google Scholar

[39]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, $3^{rd}$ edition, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar

[40]

B. Rüdiger, Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces, Stoch. Stoch. Rep., 76 (2004), 213-242.  doi: 10.1080/10451120410001704081.  Google Scholar

[41]

L. Sundbye, Global existence for the Cauchy problem for the viscous shallow water equations, Rocky Mountain J. Math., 28 (1998), 1135-1152.  doi: 10.1216/rmjm/1181071760.  Google Scholar

[42]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, $2^{nd}$ edition, volume 66 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, first edition, 1983; second edition, 1995. doi: 10.1137/1.9781611970050.  Google Scholar

[43]

Roger Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer Science & Business Media, Vol. 68, 2012. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[44]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[45]

S. ThualA. J. MajdaN. Chen and S. N. Stechmann, Simple stochastic model for El Niño with westerly wind bursts, P. Natl. Acad. Sci. USA, 113 (2016), 10245-10250.  doi: 10.1073/pnas.1612002113.  Google Scholar

[46]

K. R. Tubbs and F. T.-C. Tsai, GPU accelerated lattice Boltzmann model for shallow water flow and mass transport, Internat. J. Numer. Methods Engrg., 86 (2011), 316-334.  doi: 10.1002/nme.3066.  Google Scholar

[47]

A. Truman and J.-L Wu, Stochastic Burgers equation with Lévy space-time white noise, in, Probabilistic Methods in Fluids doi: 10.1142/9789812703989_0020.  Google Scholar

[48]

W. Wang and C.-J. Xu, The Cauchy problem for viscous shallow water equations, Rev. Mat. Iberoamericana, 21 (2005), 1-24.  doi: 10.4171/RMI/412.  Google Scholar

show all references

References:
[1]

D. Aldous, Stopping times and tightness, Ann. Probability, 6 (1978), 335-340.  doi: 10.1214/aop/1176995579.  Google Scholar

[2]

A. Bensoussan, Stochastic Navier-Stokes equations, Acta Appl. Math., 38 (1995), 267-304.  doi: 10.1007/BF00996149.  Google Scholar

[3]

H. BessaihE. Hausenblas and P. A. Razafimandimby, Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1661-1697.  doi: 10.1007/s00030-015-0339-9.  Google Scholar

[4]

P. Billingsley, Convergence of Probability Measures, $2^{nd}$ edition, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[5]

Z. BrzeźniakHa usenblas and J. Zhu, 2D stochastic Navier-Stokes equations driven by jump noise, Nonlinear Anal., 79 (2013), 122-139.  doi: 10.1016/j.na.2012.10.011.  Google Scholar

[6]

Z. BrzeźniakE. Hausenblas and P. A. Razafimandimby, Stochastic reaction-diffusion equations driven by jump processes, Potential Anal., 49 (2018), 131-201.  doi: 10.1007/s11118-017-9651-9.  Google Scholar

[7]

Z. Brzeniak, W. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Analysis: Real World Applications, 17 (2014), 283–310. doi: 10.1016/j.nonrwa.2013.12.005.  Google Scholar

[8]

N. Chen and A. J. Majda, Simple dynamical models capturing the key features of the Central Pacific El Niño, P. Natl. Acad. Sci. USA, 113 (2016), 11732-11737.   Google Scholar

[9]

Q. ChenC. Miao and Z. Zhang, On the well-posedness for the viscous shallow water equations, SIAM J. Math. Anal., 40 (2008), 443-474.  doi: 10.1137/060660552.  Google Scholar

[10]

J. Cyr, S. Tang and R. Temam, A comparison of two settings for stochastic integration with respect to Lévy processes in infinite dimensions, in Trends in Applications of Mathematics to Mechanics (eds. E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin), Springer INdAM Series, (2018), 289–373.  Google Scholar

[11]

Justin Cyr, Phuong Nguyen, Sisi Tang and Roger Temam, Review of local and global existence results for stochastic PDEs with Lévy noise, to appear. Google Scholar

[12]

Justin Cyr, Phuong Nguyen, Sisi Tang and Roger Temam, Stochastic one layer shallow equation with Lévy noise, Discrete and Continuous Dynamical Systems B, Special issue in honor of Peter Kloeden, 24 (2019), 3765–3818.  Google Scholar

[13]

Justin Cyr, Phuong Nguyen, Sisi Tang and Roger Temam, The Euler equations of an inviscid incompressible fluid driven a Lévy noise, Nonlinear Analysis: Real World Applications, 44 (2018), 173–222. doi: 10.1016/j.nonrwa.2018.04.002.  Google Scholar

[14]

Mickaël D. Chekroun and Lionel J. Roques, Models of population dynamics under the influence of external perturbations: mathematical results, Comptes Rendus Mathematique, 343 (2006), 307–310. doi: 10.1016/j.crma.2006.07.012.  Google Scholar

[15]

Mickaël D. Chekroun, Eunhee Park and Roger Temam, The Stampacchia maximum principle for stochastic partial differential equations and applications, J. Differential Equations, 260 (2016), 2926–2972. doi: 10.1016/j.jde.2015.10.022.  Google Scholar

[16]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, $2^{nd}$ edition, volume 152 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781107295513.  Google Scholar

[17]

D. Appblebaum and J.-L. Wu, Stochastic partial differential equations driven by Lévy space time white noise, Random Oper. Stochastic Equations, 8 (2000), 245-259.  doi: 10.1515/rose.2000.8.3.245.  Google Scholar

[18]

D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. Ⅱ: General Theory and Structure, $2^{nd}$ edition, Probability and its Applications, Springer, New York, 2008. doi: 10.1007/978-0-387-49835-5.  Google Scholar

[19]

A. DebusscheN. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Phys. D, 240 (2011), 1123-1144.  doi: 10.1016/j.physd.2011.03.009.  Google Scholar

[20]

A. DebusscheN. Glatt-HoltzR. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118.  doi: 10.1088/0951-7715/25/7/2093.  Google Scholar

[21]

B. Di MartinoF. J. Chatelon and P. Orenga, The nonlinear Galerkin method applied to shallow water equations, Math. Models Methods Appl. Sci., 9 (1999), 825-854.  doi: 10.1142/S0218202599000397.  Google Scholar

[22]

Z. Dong and Y. Xie, Global solutions of stochastic 2D Navier-Stokes equations with Lévy noise, Sci. China Ser. A, 52 (2009), 1497-1524.  doi: 10.1007/s11425-009-0124-5.  Google Scholar

[23]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.  Google Scholar

[24]

B. P. W. Fernando and S. S. Sritharan, Nonlinear filtering of stochastic Navier-Stokes equation with Itô-Lévy noise, Stoch. Anal. Appl., 31 (2013), 381-426.  doi: 10.1080/07362994.2013.759482.  Google Scholar

[25]

F. Flandoli and D. Gątarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[26]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differential Equations, 14 (2009), 567-600.   Google Scholar

[27]

I. Gyöngy and N. Krylov, Existence of strong solutions for Itô's stochastic equations via approximations, Probab. Theory Related Fields, 105 (1996), 143-158.  doi: 10.1007/BF01203833.  Google Scholar

[28]

D. J. Higham and P. E. Kloeden, Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math., 205 (2007), 949-956.  doi: 10.1016/j.cam.2006.03.039.  Google Scholar

[29]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, $2^{nd}$ edition, volume 24 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.  Google Scholar

[30]

T. G. Kurtz, The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities, Electron. J. Probab., 12 (2007), 951-965.  doi: 10.1214/EJP.v12-431.  Google Scholar

[31]

J. LinkP. Nguyen and R. Temam, Local martingale solutions to the stochastic one layer shallow water equations, J. Math. Anal. Appl., 448 (2017), 93-139.  doi: 10.1016/j.jmaa.2016.10.036.  Google Scholar

[32]

M. Métivier, Semimartingales: A Course on Stochastic Processes, volume 2 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin-New York, 1982.  Google Scholar

[33]

M. Métivier, Stochastic Partial Differential Equations in Infinite-dimensional Spaces, Scuola Normale Superiore di Pisa. Quaderni. [Publications of the Scuola Normale Superiore of Pisa]. Scuola Normale Superiore, Pisa, 1988.  Google Scholar

[34]

E. Motyl, Stochastic Navier-Stokes equations driven by Lévy noise in unbounded 3D domains, Potential Anal., 38 (2013), 863-912.  doi: 10.1007/s11118-012-9300-2.  Google Scholar

[35]

E. Motyl, Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains–abstract framework and applications, Stochastic Process. Appl., 124 (2014), 2052-2097.  doi: 10.1016/j.spa.2014.01.009.  Google Scholar

[36]

P. Orenga, Un théorème d'existence de solutions d'un problème de shallow water, Arch. Rational Mech. Anal., 130 (1995), 183-204.  doi: 10.1007/BF00375155.  Google Scholar

[37]

S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, volume 113 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.  Google Scholar

[38]

M. Röckner and T. Zhang, Stochastic evolution equations of jump type: existence, uniqueness and large deviation principles, Potential Anal., 26 (2007), 255-279.  doi: 10.1007/s11118-006-9035-z.  Google Scholar

[39]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, $3^{rd}$ edition, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar

[40]

B. Rüdiger, Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces, Stoch. Stoch. Rep., 76 (2004), 213-242.  doi: 10.1080/10451120410001704081.  Google Scholar

[41]

L. Sundbye, Global existence for the Cauchy problem for the viscous shallow water equations, Rocky Mountain J. Math., 28 (1998), 1135-1152.  doi: 10.1216/rmjm/1181071760.  Google Scholar

[42]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, $2^{nd}$ edition, volume 66 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, first edition, 1983; second edition, 1995. doi: 10.1137/1.9781611970050.  Google Scholar

[43]

Roger Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer Science & Business Media, Vol. 68, 2012. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[44]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.  Google Scholar

[45]

S. ThualA. J. MajdaN. Chen and S. N. Stechmann, Simple stochastic model for El Niño with westerly wind bursts, P. Natl. Acad. Sci. USA, 113 (2016), 10245-10250.  doi: 10.1073/pnas.1612002113.  Google Scholar

[46]

K. R. Tubbs and F. T.-C. Tsai, GPU accelerated lattice Boltzmann model for shallow water flow and mass transport, Internat. J. Numer. Methods Engrg., 86 (2011), 316-334.  doi: 10.1002/nme.3066.  Google Scholar

[47]

A. Truman and J.-L Wu, Stochastic Burgers equation with Lévy space-time white noise, in, Probabilistic Methods in Fluids doi: 10.1142/9789812703989_0020.  Google Scholar

[48]

W. Wang and C.-J. Xu, The Cauchy problem for viscous shallow water equations, Rev. Mat. Iberoamericana, 21 (2005), 1-24.  doi: 10.4171/RMI/412.  Google Scholar

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