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FRINATEK-Fri prosj.st. mat.,naturv.,tek

Stochastic Partial Differential Equations with Irregular Drift Coefficients

Alternative title: null

Awarded: NOK 3.4 mill.

Mathematical understanding and description of fluids is often referred to as one of the biggest remaining challenges in mathematics and mathematical physics. Liquids in motion can many times develop irregular behavior sometimes referred to as turbulence. In some cases this turbulence can be so difficult to describe, that from a modelling point of view it is more reasonable to assume that there is something random that is infulencing the system, which we call noise. The most popular equation modelling flow of liquids is called the Navier-Stokes equation. One focus of this project has been to study Navier-Stokes equation in 2 dimensions under the influence of noise or randomness. Comparing the macroscopic and the microscopic models for fluids it is clear that the noise in such an equation should arise in so-called transport form, i.e. multiplicative. Since the noise is very irregular it is reasonable to assume that also the solutions will behave as irregular, at least as a function of time. From a purely mathematical point of view it is not clear how to define the product of two such generalized functions as the solution and the noise turn out to be. By using recently developed theory it is fortunately possible to interpret the multiplicative terms by using so-called Rough Path theory, and in this project these formulations and techniques has been used for the first time to study Navier-Stokes equation. This formulation has the advantage that one get stability in the noise term as opposed what happens in the classical case were Itô calculus usually gives meaning to the term. Another advantage is that it is possible to consider many different types of noise in the equation, in particular noise that has memory as opposed to the usual assumption of Markov-noise.

In recent years our group has studied the regularizing effect of Brownian noise on differential equations with irregular coefficients. In particular we have proved that for Stochastic Differential Equations (SDE's) with merely bounded and measurable drift coefficient, the solution has the nice property that it is Malliavin differentiable. Moreover, we where able to prove existence of a Sobolev-differentiable stochastic flow to these equations. This project will deal with extending these result to the infi nite-dimensional setting, i.e. Stochastic Partial Differential Equations (SPDE's). A group in Italy have recently (2011) proven that there exists unique strong solutions to SPDE's driven by cylindrical Brownian noise and a bounded and measurable drift. Ho wever, this result gives a solution for almost every initial condition, and thus is dependent on a measure on the underlying infinite dimensional space. This rules out the study of stochastic flows for the equation. I aim with this project to be able to p rove that there exists a unique solution to the equations for every initial conditions, and moreover that the solutions are Malliavin differentiable. This would be a more complete generalization of the famous results by Zvonkin and Veretennikov to the inf inite dimensional case.

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FRINATEK-Fri prosj.st. mat.,naturv.,tek