2D backward stochastic Navier–Stokes equations with nonlinear forcing

Qiu, Jinniao; Tang, Shanjian; You, Yuncheng

doi:10.1016/j.spa.2011.08.010

Cited by 14 publications

(10 citation statements)

References 22 publications

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“…In the dynamic programming theory, some nonlinear BSPDEs as the backward stochastic Hamilton-Jacobi-Bellman equations, are also introduced in the investigation of non-Markovian control problems (see e.g [9,26]). Recently, there are many papers studying backward stochastic partial differential equations (see [6,28,29,36,38,42] and the references therein). In [39] a very general system of backward stochastic partial differential equations is studied, and in [29,36] the authors concentrate on the study of the backward stochastic 2D Navier-Stokes equation (BSNSE).…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

Liu

Zhu

2020

Forum Mathematicum

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In this paper we show the existence and uniqueness of strong solutions for a large class of backward SPDE where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE models.

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Section: Introductionmentioning

confidence: 99%

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

Liu

Zhu

2020

Forum Mathematicum

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“…They arise in many applications of probability theory and stochastic processes, for instance in the nonlinear filtering and stochastic control theory for processes with incomplete information, as an adjoint equation of the Duncan-Mortensen-Zakai filtration equation (for instance, see [1,17,18,39,44]). The representation relationship between forward-backward stochastic differential equations and BSPDEs yields the stochastic Feynman-Kac formula (see [17,26,37]). In addition, as the obstacle problems of BSPDEs, the reflected BSPDE arises as the HJB equation for the optimal stopping problems (see [3,28,38,43]).…”

Section: Introductionmentioning

confidence: 99%

Weak solution for a class of fully nonlinear stochastic Hamilton–Jacobi–Bellman equations

Qiu

2017

Stochastic Processes and their Applications

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This paper is concerned with the stochastic Hamilton-Jacobi-Bellman equation with controlled leading coefficients, which is a type of fully nonlinear backward stochastic partial differential equation (BSPDE for short). In order to formulate the weak solution for such kind of BSPDEs, a class of regular random parabolic potentials are introduced in the backward stochastic framework. The existence and uniqueness of weak solution is proved, and for the partially non-Markovian case, we obtain the associated gradient estimate. As a byproduct, the existence and uniqueness of solution for a class of degenerate reflected BSPDEs is discussed as well. (2010): 60H15, 49L20, 93E20, 35D30 Mathematics Subject Classification

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“…Other quite related works include Albeverio and Belopolskaya [1] who constructed a weak solution of the 3D Navier-Stokes equation by solving the associated stochastic system with the approach of stochastic flows, Cruzeiro and Shamarova [15] who established a connection between the strong solution to the spatially periodic Navier-Stokes equations and a solution to a system of FBSDEs on the group of volumepreserving diffeomorphisms of a flat torus, and Qiu, Tang and You [46] who considered a similar non-Markovian FBSDS to ours (1.7) in the two-dimensional spatially periodic case, and studied the wellposedness of the corresponding backward stochastic PDEs.…”

Section: Introductionmentioning

confidence: 99%

Forward–backward stochastic differential systems associated to Navier–Stokes equations in the whole space

Delbaen

Qiu

Tang

2015

Stochastic Processes and their Applications

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3A coupled forward-backward stochastic differential system (FBSDS) is formulated in spaces of fields 4 for the incompressible Navier-Stokes equation in the whole space. It is shown to have a unique local solu-5 tion, and further if either the Reynolds number is small or the dimension of the forward stochastic differ-6 ential equation is equal to two, it can be shown to have a unique global solution. These results are shown 7 with probabilistic arguments to imply the known existence and uniqueness results for the Navier-Stokes 8 equation, and thus provide probabilistic formulas to the latter. Related results and the maximum principle 9 are also addressed for partial differential equations (PDEs) of Burgers' type. Moreover, from truncating 10 the time interval of the above FBSDS, approximate solution is derived for the Navier-Stokes equation by 11 a new class of FBSDSs and their associated PDEs; our probabilistic formula is also bridged to the prob-12 abilistic Lagrangian representations for the velocity field, given by Constantin and Iyer (2008) and Zhang 13 ✩

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2D backward stochastic Navier–Stokes equations with nonlinear forcing

Cited by 14 publications

References 22 publications

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

Weak solution for a class of fully nonlinear stochastic Hamilton–Jacobi–Bellman equations

Forward–backward stochastic differential systems associated to Navier–Stokes equations in the whole space

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