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A priori LIpschitz estimates for unbounded solutions of local and nonlocal Hamilton-Jacobi viscous equations with Ornstein-Uhlenbeck Operator Ley, Olivier


In this work, in collaboration with Emmanuel Chasseigne (Tours) and Thi Tuyen Nguyen (Rennes), we establish a priori Lipschitz estimates for unbounded solutions of viscous Hamilton-Jacobi equations in presence of a Ornstein-Uhlenbeck drift. The first type of equations we consider are local. The Ornstein-Uhlenbeck drift is associated with a general diffusion operator. This part is a generalization of an earlier work of Fujita, Ishii & Loreti (2006). The second type of equations we deal with are nonlocal. The Ornstein-Uhlenbeck term is associated with a integro-differential operator of Fractional Laplacian type. In both case, we obtain some local Lipschitz estimates which are independent of the L^\infty norm of the solution (which is supposed to have at most an exponential growth). These results can be applied to prove the large time behavior of the solutions.

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