Nontrivial Steady State and Non-Uniqueness for the Navier-Stokes Equations




RE 104

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Alexey Cheskidov

Department of Mathematics, University of Illinois at Chicago




We prove that there exists a nontrivial finite energy periodic stationary weak solution to the 3D Navier-Stokes equations (NSE). The construction relies on a convex integration scheme utilizing new stationary building blocks designed specifically for the NSE. It provides the first proof of a non-uniqueness for the stationary Naiver-Stokes equations. Moreover, the result gives an alternative proof of a non-uniqueness for the evolutionary Naiver-Stokes equations, recently obtained by Buckmaster and Vicol. Indeed, a nontrivial stationary solution can be used as an initial value for the evolutionary problem. Leray’s theorem implies the existence of a Leray-Hopf solution starting from this initial data, which cannot coincide with the constructed stationary solution. The constructed family of approximate stationary solutions is also used to prove the existence of weak solutions of the NSE with energy profiles discontinuous on a dense set of positive Lebesgue measure. This is a joint work with Xiaoyutao Luo.


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