# American Institute of Mathematical Sciences

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On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition
November  2010, 27(4): 1375-1389. doi: 10.3934/dcds.2010.27.1375

## Systems of Bellman equations to stochastic differential games with non-compact coupling

 1 School of Management, International Center for Decision and Risk Analysis, University of Texas at Dallas, 800 W. Campbell Rd, SM30, Richardson, TX 75080-3021, United States 2 Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, NRW, 53115 Bonn, Germany 3 Harvard Mathematics Department, One Oxford Street, Cambridge, MA 02138, United States

Received  November 2009 Revised  February 2010 Published  March 2010

We consider a class of non-linear partial differential systems like

-div$(a(x)\nabla u_{\nu}) +\lambda u_{\nu}=H_{\nu}(x, Du) \,$

with applications for the solution of stochastic differential games with $N$ players, where $N$ is an arbitrary but positive number. The Hamiltonian $H$ of the non-linear system satisfies a quadratic growth condition in $D u$ and contains interactions between the players in the form of non-compact coupling terms $\nabla u_{i} \cdot\nabla u_j$. A $L^{\infty}\cap H^1$-estimate and regularity results are shown, mainly in two-dimensional space. The coupling arises from cyclic non-market interaction of the control variables.

Citation: Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375
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