Abstract
The hydrodynamical model of the collective behavior of animals consists of the Euler equation with additional non-local forcing terms representing the repulsive and attractive forces among individuals. This paper deals with the system endowed with an additional white-noise forcing and an artificial viscous term. We provide a proof of the existence of a dissipative martingale solution—a cornerstone for a subsequent analysis of the system with stochastic forcing.
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For a function \(f:{\mathbb {T}}\mapsto {\mathbb {R}}\), we define its average as \((f)_{\mathbb {T}}:= \int _{\mathbb {T}}f \ \mathrm{d}x\).
Here we adopt the following notation. For a general function \(f(\varrho _\varepsilon ,\mathbf{u}_\varepsilon )\), we use \(\overline{f(\varrho ,\mathbf{u})}\) to denote its weak limit (which is generaly not equal to \(f(\varrho ,\mathbf{u})\)).
References
Bensoussan, A., Temam, R.: Équations stochastiques du type Navier–Stokes. J. Funct. Anal. 13, 195–222 (1973)
Breit, D.: An introduction to stochastic Navier–Stokes equations. In: Bulícek, M., Feireisl, E., Pokorný, M. (eds.) New Trends and Results in Mathematical Description of Fluid Flows. Necčas Center Series, pp. 1–51. Springer, Cham (2018)
Breit, D., Feireisl, E., Hofmanová, M.: Compressible fluids driven by stochastic forcing: the relative energy inequality and applications. Commun. Math. Phys. 350(2), 443–473 (2017)
Breit, D., Feireisl, E., Hofmanová, M.: Stochastically Forced Compressible Fluid Flows. De Gruyter Series in Applied and Numerical Mathematics, vol. 3. De Gruyter, Berlin (2018)
Breit, D., Hofmanová, M.: Stochastic Navier–Stokes equations for compressible fluids. Indiana Univ. Math. J. 65(4), 1183–1250 (2016)
Brzeźniak, Z., Ondreját, M., Seidler, J.: Invariant measures for stochastic nonlinear beam and wave equations. J. Differ. Equ. 260(5), 4157–4179 (2016)
Březina, J., Mácha, V.: Inviscid limit for the compressible Euler system with non-local interactions. J. Differ. Equ. 267(7), 4410–4428 (2019)
Cañizo, J.A., Carrillo, J.A., Rosado, J.: A well-posedness theory in measures for some kinetic models of collective motion. Math. Models Methods Appl. Sci. 21(3), 515–539 (2011)
Carrillo, J.A., Feireisl, E., Gwiazda, P., Gwiazda, A.Ś: Weak solutions for Euler systems with non-local interactions. J. Lond. Math. Soc. (2) 95(3), 705–724 (2017)
Chiodaroli, E., Kreml, O., Mácha, V., Schwarzacher, S.: Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data. Trans. Am. Math. Soc. 374(4), 2269–2295 (2021)
Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Trans. Autom. Control 52(5), 852–862 (2007)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)
Davidson, D.: Actions, reasons, and causes. J. Philos. 60, 685–700 (1963)
De Lellis, C., Székelyhidi, L., Jr.: The Euler equations as a differential inclusion. Ann. Math. (2) 170(3), 1417–1436 (2009)
Debussche, A., Glatt-Holtz, N., Temam, R.: Local martingale and pathwise solutions for an abstract fluids model. Phys. D 240(14–15), 1123–1144 (2011)
Denk, R., Hieber, M., Prüss, J.: R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166(788), viii+114 (2003)
Dennett, D.: Elbow Room: The Varieties of Free Will Worth Wanting. MIT Press, Cambridge (1984)
Diening, L., Růžička, M., Schumacher, K.: A decomposition technique for John domains. Ann. Acad. Sci. Fenn. Math. 35(1), 87–114 (2010)
Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and its Applications, vol. 26. Oxford University Press, Oxford (2004)
Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids. Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel (2009)
Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001)
Jakubowski, A.: The almost sure skorokhod representation for subsequences in nonmetric spaces. Teoriya Veroyatnostei i ee Primeneniya 42, 209–216 (1997)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991)
Karper, T.K., Mellet, A., Trivisa, K.: Existence of weak solutions to kinetic flocking models. SIAM J. Math. Anal. 45(1), 215–243 (2013)
Levy, N., McKenna, M.: Recent work on free will and moral responsibility. Philos. Compass 43, 96–133 (2008)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel (1995). [2013 reprint of the 1995 original] [MR1329547]
Marinelli, C., Röckner, M.: On the maximal inequalities of Burkholder, Davis and Gundy. Expos. Math. 34(1), 1–26 (2016)
Novotný, A., Straškraba, I.: Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Series in Mathematics and its Applications, vol. 27. Oxford University Press, Oxford (2004)
Strawson, P.F.: Freedom and resentment. Proc. Br. Acad. 48, 187–211 (1962)
Acknowledgements
The work of Václav Mácha was supported by the Czech Science Foundation (GAČR), Grant Agreement GA18-05974S in the framework of RVO:67985840. The work of Pavel Ludvík was supported by the Grant IGA_PrF_2021_008 “Mathematical Models” of the Internal Grant Agency of Palacký University in Olomouc.
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Ludvík, P., Mácha, V. Stochastic Forcing in Hydrodynamic Models with Non-local Interactions. J Theor Probab 35, 2806–2852 (2022). https://doi.org/10.1007/s10959-021-01137-x
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DOI: https://doi.org/10.1007/s10959-021-01137-x