Theory of Probability and Mathematical Statistics
Initial-boundary value problem for transport equations driven by rough paths
Dai Noboriguchi
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Abstract: In this paper, we are interested in the initial Dirichlet boundary value problem for a transport equation driven by weak geometric Hölder p-rough paths. We introduce a notion of solutions to rough partial differential equations with boundary conditions. Consequently, we will establish a well-posedness for such a solution under some assumptions stated below. Moreover, the solution is given explicitly.
Keywords: Initial-boundary value problem, transport equation, rough paths
Bibliography: Herbert Amann, Ordinary differential equations, De Gruyter Studies in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1990. An introduction to nonlinear analysis; Translated from the German by Gerhard Metzen. MR 1071170, DOI 10.1515/9783110853698
L. Arlotti, J. Banasiak, and B. Lods, Semigroups for general transport equations with abstract boundary conditions, arXiv:math.AP/0610808.
Luigi Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158 (2004), no. 2, 227–260. MR 2096794, DOI 10.1007/s00222-004-0367-2
Claude Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport, Ann. Sci. École Norm. Sup. (4) 3 (1970), 185–233 (French). MR 274925, DOI 10.24033/asens.1190
Franck Boyer, Trace theorems and spatial continuity properties for the solutions of the transport equation, Differential Integral Equations 18 (2005), no. 8, 891–934. MR 2150445
Michael Caruana and Peter Friz, Partial differential equations driven by rough paths, J. Differential Equations 247 (2009), no. 1, 140–173. MR 2510132, DOI 10.1016/j.jde.2009.01.026
Gianluca Crippa, Carlotta Donadello, and Laura V. Spinolo, Initial-boundary value problems for continuity equations with BV coefficients, J. Math. Pures Appl. (9) 102 (2014), no. 1, 79–98 (English, with English and French summaries). MR 3212249, DOI 10.1016/j.matpur.2013.11.002
A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing, J. Funct. Anal. 259 (2010), no. 4, 1014–1042. MR 2652180, DOI 10.1016/j.jfa.2010.02.016
R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547. MR 1022305, DOI 10.1007/BF01393835
F. Flandoli, M. Gubinelli, and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math. 180 (2010), no. 1, 1–53. MR 2593276, DOI 10.1007/s00222-009-0224-4
Peter K. Friz and Nicolas B. Victoir, Multidimensional stochastic processes as rough paths, Cambridge Studies in Advanced Mathematics, vol. 120, Cambridge University Press, Cambridge, 2010. Theory and applications. MR 2604669, DOI 10.1017/CBO9780511845079
Tadahisa Funaki, Construction of a solution of random transport equation with boundary condition, J. Math. Soc. Japan 31 (1979), no. 4, 719–744. MR 544688, DOI 10.2969/jmsj/03140719
Martina Hofmanová, Scalar conservation laws with rough flux and stochastic forcing, Stoch. Partial Differ. Equ. Anal. Comput. 4 (2016), no. 3, 635–690. MR 3538012, DOI 10.1007/s40072-016-0072-3
Martina Hofmanová, A Bhatnagar-Gross-Krook approximation to stochastic scalar conservation laws, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 4, 1500–1528 (English, with English and French summaries). MR 3414456, DOI 10.1214/14-AIHP610
C. Imbert and J. Vovelle, A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications, SIAM J. Math. Anal. 36 (2004), no. 1, 214–232. MR 2083859, DOI 10.1137/S003614100342468X
Kazuo Kobayasi and Dai Noboriguchi, Well-posedness for stochastic scalar conservation laws with the initial-boundary condition, J. Math. Anal. Appl. 461 (2018), no. 2, 1416–1458. MR 3765499, DOI 10.1016/j.jmaa.2018.01.054
Kazuo Kobayasi and Dai Noboriguchi, A stochastic conservation law with nonhomogeneous Dirichlet boundary conditions, Acta Math. Vietnam. 41 (2016), no. 4, 607–632. MR 3574057, DOI 10.1007/s40306-015-0157-5
Hiroshi Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1990. MR 1070361
Pierre-Louis Lions, Benoît Perthame, and Panagiotis E. Souganidis, Scalar conservation laws with rough (stochastic) fluxes, Stoch. Partial Differ. Equ. Anal. Comput. 1 (2013), no. 4, 664–686. MR 3327520, DOI 10.1007/s40072-013-0021-3
Pierre-Louis Lions, Benoît Perthame, and Panagiotis E. Souganidis, Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case, Stoch. Partial Differ. Equ. Anal. Comput. 2 (2014), no. 4, 517–538. MR 3274890, DOI 10.1007/s40072-014-0038-2
Terry Lyons and Zhongmin Qian, System control and rough paths, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2002. Oxford Science Publications. MR 2036784, DOI 10.1093/acprof:oso/9780198506485.001.0001
Stéphane Mischler, On the trace problem for solutions of the Vlasov equation, Comm. Partial Differential Equations 25 (2000), no. 7-8, 1415–1443. MR 1765137, DOI 10.1080/03605300008821554
Wladimir Neves and Christian Olivera, Initial-boundary value problem for stochastic transport equations, Stoch. Partial Differ. Equ. Anal. Comput. 9 (2021), no. 3, 674–701. MR 4297236, DOI 10.1007/s40072-020-00180-9
