In this work we prove the existence and uniqueness of pathwise solutions up to a stopping time to the stochastic Euler equations perturbed by additive and multiplicative Lévy noise in two and three dimensions. The existence of a unique maximal solution is also proved.
D.Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, Vol. 93, Cambridge University Press, 2004.
2.
H.Bessaih and F.Flandoli, 2-D Euler equation perturbed by noise, Nonlinear Differential Equations and Applications6 (1998), 35–54. doi:10.1007/s000300050063.
3.
H.Bessaih, E.Hausenblas and P.Razafimandimby, Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type, Nonlinear Differential Equations and Applications22 (2015), 1661–1697. doi:10.1007/s00030-015-0339-9.
4.
H.Bessaih, Martingale solutions for stochastic Euler equations, Stochastic Analysis and Applications17(5) (1999), 713–725. doi:10.1080/07362999908809631.
5.
J.P.Bourguignon and H.Brezis, Remarks on the Euler equation, Journal of Functional Analysis15 (1974), 341–363. doi:10.1016/0022-1236(74)90027-5.
6.
Z.Brzeźniak, E.Hausenblas and P.Razafimandimby, Stochastic non-parabolic dissipative systems modeling the flow of liquid crystals: Strong solution, Mathematical Analysis of Incompressible Flow1875 (2014), 41–72.
Z.Brzeźniak and S.Peszat, Stochastic two dimensional Euler equations, The Annals of Probability29(4) (2001), 1796–1832. doi:10.1214/aop/1015345773.
9.
M.Cannone, Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations, Handbook of Mathematical Fluid Mechanics, Vol. III, Elsevier, 2004.
10.
N.Capinski and N.Cutland, Stochastic Euler equations on the torus, Annals of Applied Probability9 (1998), 688–705. doi:10.1214/aoap/1029962809.
11.
J.-Y.Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and Its Applications, Vol. 14, The Clarendon Press, Oxford University Press, New York, 1998.
12.
M.Christ, Lectures on Singular Integral Operators, American Mathematical Society, Providence, Rhode Island, 1990.
13.
P.Constantin, On the Euler equations of incompressible fluids, Bulletin of the American Mathematical Society44(4) (2007), 603–621. doi:10.1090/S0273-0979-07-01184-6.
14.
A.Cruzeiro, F.Flandoli and P.Malliavin, Brownian motion on volume preserving diffeomorphisms group and existence of global solutions of 2-D stochastic Euler equation, Journal of Functional Analysis242(1) (2007), 304–326. doi:10.1016/j.jfa.2006.06.010.
15.
A.Cruzeiro and I.Torrecilla, On a 2-D stochastic Euler equation of transport type: Existence and geometric formulation, Stochastics and Dynamics15(1) (2015), 1–19. doi:10.1142/S0219493714500129.
16.
G.Da Prato and J.Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992.
17.
C.L.Fefferman, D.S.McCormick, J.C.Robinson and J.L.Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, Journal of Functional Analysis267 (2014), 1035–1056. doi:10.1016/j.jfa.2014.03.021.
18.
B.P.W.Fernando, B.Rüdiger and S.S.Sritharan, Mild solutions of stochastic Navier–Stokes equations with jump noise in -spaces, Mathematische Nachrichten288(14–15) (2015), 1615–1621. doi:10.1002/mana.201300248.
19.
B.P.W.Fernando and S.S.Sritharan, Nonlinear filtering of stochastic Navier–Stokes equation with Lévy noise, Stochastic Analysis and Applications31 (2013), 1–46. doi:10.1080/07362994.2012.727144.
20.
N.Glatt-Holtz and V.C.Vicol, Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, The Annals of Probability42(1) (2014), 80–145. doi:10.1214/12-AOP773.
21.
L.Hörmander, The Analysis of Linear Partial Differential Operators III, Pseudo-Differential Operators, Springer-Verlag, Berlin, Heidelberg, 1994.
22.
T.Kato, Nonlinear evolution equations and the Euler flow, Journal of Functional Analysis56 (1984), 15–28. doi:10.1016/0022-1236(84)90024-7.
23.
T.Kato and G.Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Communications in Pure and Applied Mathematics41(7) (1988), 891–907. doi:10.1002/cpa.3160410704.
24.
J.U.Kim, On the stochastic Euler equation in a two-dimensional domain, SIAM Journal on Mathematical Analysis33(5) (2002), 1211–1227. doi:10.1137/S0036141001383941.
25.
J.U.Kim, Existence of a local smooth solution in probability to the stochastic Euler equations in , Journal of Functional Analysis256 (2009), 3660–3687. doi:10.1016/j.jfa.2009.03.012.
26.
J.U.Kim, On the stochastic quasi-linear symmetric hyperbolic system, Journal of Differential Equations250 (2011), 1650–1684. doi:10.1016/j.jde.2010.09.025.
27.
J.L.Lions and E.Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York, 1972.
28.
A.J.Majda and A.L.Bertozzi, Vorticity and Incompressible Flow, Cambridge Text Appl. Math., Vol. 27, Cambridge University Press, Cambridge, 2002.
29.
V.Mandrekar and B.Rüdiger, Existence and uniqueness of path-wise solutions for stochastic integral equations driven by Lévy noise on separable Banach space, Stochastics78(4) (2006), 189–212.
30.
V.Mandrekar and B.Rüdiger, Stochastic Integration in Banach Spaces: Theory and Applications, Springer, Cham, Heidelberg, New York, 2015.
31.
C.Marinelli and M.Röckner, On the maximal inequalities of Burkholder, Davis and Gundy, Expositiones Mathematicae, 2015, Article in Press.
32.
M.Métivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces, Publications of the Scuola Normale Superiore, Quaderni, Pisa, 1988.
33.
R.Mikulevičius and G.Valiukevičius, On stochastic Euler equations, Lithuanian Mathematical Journal2 (1998), 181–192. doi:10.1007/BF02465554.
34.
R.Mikulevičius and G.Valiukevičius, On stochastic Euler equation in , Electronic Journal of Probability5(6) (2000), 1–20.
35.
M.T.Mohan and S.S.Sritharan, New methods for local solvability of quasilinear symmetric hyperbolic systems, Evolution Equations and Control Theory5(2) (2016), 273–302. doi:10.3934/eect.2016005.
36.
M.T.Mohan and S.S.Sritharan, -solutions of the stochastic Navier–Stokes equations subject to Lévy noise with initial data, Submitted for Journal Publication.
37.
E.Motyl, Stohastic Navier–Stokes equations driven by Lévy noise in unbounded 3D domains, Potential Analysis38(3) (2013), 863–912.
38.
N.S.Papageorgiou and S.Kyritsi-Yiallourou, Handbook of Applied Analysis, 2nd edn, Springer-Verlag, New York, 2009.
39.
P.E.Protter, Stochastic Integration and Differential Equations, 2nd edn, Springer-Verlag, New York, 2005.
40.
J.C.Robinson, Infinite-Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, UK, 2001.
41.
B.Rüdiger and G.Ziglio, Itô formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces, Stochastics78(6) (2006), 377–410.
42.
K.Sakthivel and S.S.Sritharan, Martingale solutions for stochastic Navier–Stokes equations driven by Lévy noise, Evolution Equations and Control Theory1 (2012), 355–392. doi:10.3934/eect.2012.1.355.
43.
M.A.Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, Heidelberg, 2001.
44.
E.M.Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970.
45.
R.Temam, On the Euler equations of incompressible perfect fluids, Journal of Functional Analysis20(6) (1975), 32–43. doi:10.1016/0022-1236(75)90052-X.
46.
R.Temam, Local existence of C∞ solutions of the Euler equations of incompressible perfect fluids, in: Lecture Notes in Mathematics, Vol. 565, Springer-Verlag, New York, 1976, pp. 184–194.
