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Non-uniqueness of Leray solutions of the forced Navier-Stokes equations. (English) Zbl 1497.35337

Summary: In a seminal work, J. Leray [Acta Math. 63, 193–248 (1934; JFM 60.0726.05)] demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a “background” solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional vortex constructed by M. Vishik [“Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. I”, Preprint, arXiv:1805.09426; “Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. II”, Preprint, arXiv:1805.09440]. The second solution is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of H. Jia and V. Sverak [J. Funct. Anal. 268, No. 12, 3734–3766 (2015; Zbl 1317.35176)]. Our solutions live precisely on the borderline of the known well-posedness theory.

MSC:

35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35D30 Weak solutions to PDEs
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
76D05 Navier-Stokes equations for incompressible viscous fluids

References:

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