An energy method for rough partial differential equations

Hocquet A, Hofmanová M (2018)
JOURNAL OF DIFFERENTIAL EQUATIONS 265(4): 1407-1466.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Hocquet, Antoine; Hofmanová, MartinaUniBi
Abstract / Bemerkung
We present a well-posedness and stability result for a class of nondegenerate linear parabolic equations driven by geometric rough paths. More precisely, we introduce a notion of weak solution that satisfies an intrinsic formulation of the equation in a suitable Sobolev space of negative order. Weak solutions are then shown to satisfy the corresponding energy estimates which are deduced directly from the equation. Existence is obtained by showing compactness of a suitable sequence of approximate solutions whereas uniqueness relies on a doubling of variables argument and a careful analysis of the passage to the diagonal. Our result is optimal in the sense that the assumptions on the deterministic part of the equation as well as the initial condition are the same as in the classical PDEs theory. (C) 2018 Elsevier Inc. All rights reserved.
Stichworte
Rough paths; Rough PDEs; Energy method; Weak solutions
Erscheinungsjahr
2018
Zeitschriftentitel
JOURNAL OF DIFFERENTIAL EQUATIONS
Band
265
Ausgabe
4
Seite(n)
1407-1466
ISSN
0022-0396
eISSN
1090-2732
Page URI
https://pub.uni-bielefeld.de/record/2920675

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Hocquet A, Hofmanová M. An energy method for rough partial differential equations. JOURNAL OF DIFFERENTIAL EQUATIONS. 2018;265(4):1407-1466.
Hocquet, A., & Hofmanová, M. (2018). An energy method for rough partial differential equations. JOURNAL OF DIFFERENTIAL EQUATIONS, 265(4), 1407-1466. doi:10.1016/j.jde.2018.04.006
Hocquet, A., and Hofmanová, M. (2018). An energy method for rough partial differential equations. JOURNAL OF DIFFERENTIAL EQUATIONS 265, 1407-1466.
Hocquet, A., & Hofmanová, M., 2018. An energy method for rough partial differential equations. JOURNAL OF DIFFERENTIAL EQUATIONS, 265(4), p 1407-1466.
A. Hocquet and M. Hofmanová, “An energy method for rough partial differential equations”, JOURNAL OF DIFFERENTIAL EQUATIONS, vol. 265, 2018, pp. 1407-1466.
Hocquet, A., Hofmanová, M.: An energy method for rough partial differential equations. JOURNAL OF DIFFERENTIAL EQUATIONS. 265, 1407-1466 (2018).
Hocquet, Antoine, and Hofmanová, Martina. “An energy method for rough partial differential equations”. JOURNAL OF DIFFERENTIAL EQUATIONS 265.4 (2018): 1407-1466.
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