Initial-boundary value problem for transport equations driven by rough paths

Author:
Dai Noboriguchi

Journal:
Theor. Probability and Math. Statist. **110** (2024), 167-183

MSC (2020):
Primary 35L04, 60H15

DOI:
https://doi.org/10.1090/tpms/1212

Published electronically:
May 10, 2024

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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.

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References
- H. Amann,
*Ordinary differential equations. An introduction to nonlinear analysis*, W. de Gruyter, Berlin, 1990. MR **1071170**
- L. Arlotti, J. Banasiak, and B. Lods,
*Semigroups for general transport equations with abstract boundary conditions*, arXiv:math.AP/0610808.
- L. Ambrosio,
*Transport equation and Cauchy problem for BV vector fields*, Invent. Math. **158** (2004), 227–260. MR **2096794**
- C. 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. MR **274925**
- F. 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**
- M. Caruana and P. Friz,
*Partial differential equations driven by rough paths*, J. Differential Equations **247** (2009), no. 1, 140–173. MR **2510132**
- G. Crippa, C. Donadello, and V. Spinolo,
*Initial-boundary value problems for continuity equations with BV coefficients*, arXiv:1304.0975v1. MR **3212249**
- A. Debussche and J. Vovelle,
*Scalar conservation laws with stochastic forcing*, J. Funct. Anal. **259** (2010), 1014–1042. MR **2652180**
- R. DiPerna and P.-L. Lions,
*Ordinary differential equations, transport theory and Sobolev spaces*, Invent. Math. **98** (1989), 511–547. MR **1022305**
- 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**
- P. Friz and N. Victoir,
*Multidimensional stochastic processes as rough paths theory and applications*, Cambridge University Press, 2010. MR **2604669**
- T. Funaki,
*Construction of a solution of random transport equation with boundary condition*, J. Math. Soc. Japan **31** (1979), no. 4, 719–744. MR **544688**
- M. Hofmanová,
*Scalar conservation laws with rough flux and stochastic forcing*, Stoch. Partial Differ. Equ. Anal. Comput. **4** (2016), 635–690. MR **3538012**
- M. Hofmanová,
*A Bhatnagar–Gross–Kross approximation to stochastic scalar conservation laws*, Ann. Inst. Henri Poincaré Probab. Stat. **51** (2015), no. 4, 1500–1528. MR **3414456**
- C. Imbert and J. Vovelle,
*A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications*, SIAM J. Math. Anal. **36** (2004), 214–232. MR **2083859**
- K. Kobayasi and D. Noboriguchi,
*Well-posedness for stochastic scalar conservation laws with the initial-boundary condition*, J. Math. Anal. Appl. **461** (2018), 1416–1458. MR **3765499**
- K. Kobayasi and D. Noboriguchi,
*A stochastic conservation law with nonhomogeneous Dirichlet boundary conditions*, Acta Math. Vietnam. **41** (2016), no. 4, 607–632. MR **3574057**
- H. Kunita,
*Stochastic flows and stochastic differential equations*, Cambridge University Press, 1990. MR **1070361**
- P.-L. Lions, B. Perthame, and P. E. Souganidis,
*Scalar conservation laws with rough (stochastic) fluxes*, Stoch. Partial Differ. Equ. Anal. Comput. **1** (2013), no. 4, 664–686. MR **3327520**
- P.-L. Lions, B. Perthame, and P. 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**
- T. J. Lyons and Z. Qian,
*System control and rough paths*, Oxford University Press, 2002. MR **2036784**
- S. Mischler,
*On the trace problem for the solutions of the Vlasov equations*, Comm. Part. Diff. Eq. **25** (2000), no. 7-8, 1415–1443. MR **1765137**
- W. Neves and C. Olivera,
*Initial-boundary value problem for stochastic transport equations*, Stoch. Partial Differ. Equ. Anal. Comput. **9** (2021), no. 3, 674–701. MR **4297236**

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Additional Information

**Dai Noboriguchi**

Affiliation:
Waseda University Senior High School, 3-31-1 Kamishakujii, Nerima-ku, Tokyo, 177-0044, Japan

Email:
nobo@waseda.jp

Keywords:
Initial-boundary value problem,
transport equation,
rough paths

Received by editor(s):
August 5, 2022

Accepted for publication:
March 7, 2023

Published electronically:
May 10, 2024

Additional Notes:
The author was supported by Waseda University Grant for Special Research Projects (No. 2021C-361 and No. 2022C-286).

Article copyright:
© Copyright 2024
Taras Shevchenko National University of Kyiv