STICKS - Stochastic Mikado Flows for Fluid Mechanics
Final Report Abstract
The main goal of this project, which was carried out at New York University’s Courant Institute of Mathematical Sciences in the group of Prof. Vlad Vicol, was to study nonuniqueness for equations of stochastic fluid dynamics. This class of equations presents an interesting test case for recent, deterministic results in nonuniqueness as noise often has a regularising effect and, in a sense, it is often easier for a solution to an equation with noise to be unique, when compared with the equation without noise. The equations studied are relevant in fluid dynamics, climate science, engineering and many adjacent fields of science and technology. The method used in this project to prove nonuniqueness is called convex integration. It has its origins in differential geometry (in the works of Nash and Gromov, among others), and it has been applied to enormous success in (deterministic) fluid dynamics within the last 15 years, most notably by De Lellis, Székelyhidi as well as Buckmaster and Vicol. The method constructs solutions to the nonlinear equations as sums of building blocks of rapidly increasing frequency (and decreasing amplitude) that are chosen in a geometrically correct way to deal with the nonlinearity. More recently (within the last roughly 5 years), this method has been applied by several people (pioneered by Hofmanová, Zhu and Zhu and others) to equations perturbed by noise. As mentioned above, these equations tend to have stronger uniqueness results, so one expects nonuniqueness to be more difficult to prove. One of the objectives of the project was to explore the tensions between these two phenomena (nonuniqueness via convex integration, uniqueness via noise), and ideally to map out the boundaries of the “unique” and “non-unique” regime of these techniques. The first result, together with Stefano Modena, established the so far most complete “mapping” of such a boundary for the simpler case of the (stochastic) transport equation (perturbed by transport noise), a first-order linear partial differential equation. We managed to complement the literature by exploring a range of parameters where nonuniqueness can be proven using the method of convex integration. The second result, in cooperation with Marco Rehmeier and Theresa Lange, concerns the fractional Navier–Stokes equations perturbed by transport noise for very small fractional exponents. Here we could show that nonuniqueness results can be achieved even in the class of Leray–Hopf solutions (as opposed to “merely” weak solutions). In order prove this, we needed to track very carefully the influence of the energy in the proof as well as a few new, technical problems. This is the first nonuniqueness result in this setting.
Publications
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Local Nonuniqueness for Stochastic Transport Equations with Deterministic Drift. SIAM Journal on Mathematical Analysis, 56(4), 5209-5261.
Modena, Stefano & Schenke, Andre
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Non-uniqueness of Leray–Hopf solutions for the 3D fractional Navier–Stokes equations perturbed by transport noise
Theresa Lange, Marco Rehmeier & Andre Schenke
