Theory of random evolutions with applications to partial differential equations

Linear Partial Differential Equations with Random Forcing

where (0 l ) is the ensemble-averaging operation. The higher order moments can be determined by forming the ensemble average of different combinations of u(x, t) with the help of (1.4). Unfortunately, the Green’s function, G(x, t ; x’, t’), of (l.l), (1.2) and (1.3) cannot be obtained in terms of elementary or special functions except for the simplest cases. In general, we will have to obtain G...

The Basic Theory for Partial Functional Differential Equations and Applications

Let (X, |·|) be an infinite dimensional Banach space and L(X) be the space of bounded linear operators from X into X. Suppose that r > 0 is a given real number. C ([−r, 0] , X) denotes the space of continuous functions from [−r, 0] to X with the uniform convergence topology and we will use simply CX for C ([−r, 0] , X). For u ∈ C ([−r, b] , X), b > 0 and t ∈ [0, b], let ut denote the element of...

Partial Differential Equations and Applications

Random fractional functional differential equations

In this paper, we prove the existence and uniqueness results to the random fractional functional differential equations under assumptions more general than the Lipschitz type condition. Moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.

Multilinear Littlewood-Paley estimates with applications to partial differential equations.

We obtain a collection of multilinear Littlewood-Paley estimates, which we then apply to two problems in partial differential equations. The first problem is the estimation of the square root of an elliptic operator in divergence form, and the second is the estimation of solutions to the Cauchy problem for nondivergence-form parabolic equations.