On the uniqueness of the flow generated by an irregular vector field

Abstract

The flow in time of an initial state ensemble in a multidimensional phase space, as a rule, models some dynamic process. Under what conditions is such a flow generated by a vector field in such a way that the given flow corresponds to the vector field in a unique way? A positive answer to this question is given by the classical uniqueness theorems for the solution of the initial value problem in the case of a regular vector field with the required properties of the modulus of continuity in space variables. In mathematical models of stochastic differential equations, in models of irregular hydrodynamic flows, and in a number of other cases when the flow is generated by a “bad” vector field that has a modulus of continuity in space variables that does not meet the conditions of the uniqueness theorem for solving the initial problem for a vector field, generating this flow, we cannot speak about the correctness of the initial problem for the vector field and, thus, about the correctness of finding the trajectories connecting the initial and actual states of the ensemble of particles in the phase space. In this case, the uniqueness of the flow generated by the vector field remains to be judged only by the properties of the flow itself. The only known result of this type is van Kampen's theorem, which states that the uniqueness of a flow generated by a vector field continuous in space variables is guaranteed by the properties of homeomorphism and the Lipschitz property of the flow in space variables. If the vector velocity field loses the property of continuity in space variables, then van Kampen's theorem does not work and some other properties of the flow are required to guarantee its uniqueness. In this paper, we establish such properties of a flow that guarantee its uniqueness even in the case of a violation of the continuity of the vector field that generates this flow. The conditions of van Kampen's theorem in a certain sense are a special case of the properties of the flow established in this paper, which guarantee its uniqueness as a solution to the initial problem for an irregular vector field. The general construction constructed here makes it possible to establish such properties of flows in various mathematical models that guarantee its uniqueness for a generating vector field.