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For the incompressible Navier-Stokes equations in R3 with low viscosity ν > 0, we consider the Cauchy problem with initial vorticity ω0 that represents an infinitely thin vortex filament of arbitrary given strength \Gamma supported on a circle. The vorticity field ω(x, t) of the solution is smooth at any positive time and corresponds to a vortex ring of thickness√νt that is translated along its symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quantified by Kelvin in 1867. For small viscosities, we show that ω(x, t) is well-approximated on a large time interval by ω_lin (x − a(t), t), where ω_lin(·, t) = exp(νt\Delta)ω0 is the solution of the heat equation with initial data ω0, and ˙a(t) is the instantaneous velocity given by Kelvin’s formula. This gives a rigorous justification of the binormal motion for circular vortex filaments in weakly viscous fluids. The proof relies on the construction of a precise approximate solution, using a perturbative expansion in self-similar variables. To verify the stability of this approximation, one needs to rule out potential instabilities coming from very large advection terms in the linearized operator. This is done by adapting V. I. Arnold’s geometric stability methods developed in the inviscid case ν = 0 to the slightly viscous situation. It turns out that although the geometric structures behind Arnold’s approach are no longer preserved by the equation for ν >0, the relevant quadratic forms behave well on larger subspaces than those originally used in Arnold’s theory and interact favorably with the viscous terms. 

We consider variational principles related to V. I. Arnold’s stability criteria for steady-state solutions of the two-dimensional incompressible Euler equation. Our goal is to investigate under which conditions the quadratic forms defined by the second variation of the associated functionals can be used in the stability analysis, both for the Euler evolution and for the Navier–Stokes equation at low viscosity. In particular, we revisit the classical example of Oseen’s vortex, providing a new stability proof with a stronger geometric flavor. Our analysis involves a fairly detailed functional-analytic study of the inviscid case, which may be of independent interest, and a careful investigation of the influence of the viscous term in the particular example of the Gaussian vortex. 

We revisit the question of whether the functions defined on the real $$m\times n$$ matrices that are convex along rank-one directions are also quasi-convex in the sense of Morrey. Using the linearity of the map $$f\to \int_{\TT^n} f(\nabla u(x))\,dx$$, we propose to study the question as a problem in convex optimization. This might be useful when trying to resolve the open cases, such as the case $m=2$, or various cases with symmetries. 

We consider the Cauchy problem for the incompressible Navier–Stokes equations in R3 for a one-parameter family of explicit scale-invariant axi-symmetric initial data, which is smooth away from the origin and invariant under the reflection with respect to the xy-plane. Working in the class of axi-symmetric fields, we calculate numerically scale-invariant solutions of the Cauchy problem in terms of their profile functions, which are smooth. The solutions are necessarily unique for small data, but for large data we observe a breaking of the reflection symmetry of the initial data through a pitchfork-type bifurcation. By a variation of previous results by Jia and ˇSver´ak (Invent Math 196(1):233–265, 2013, https://doi.org/10. 1007/s00222-013-0468-x) it is known rigorously that if the behavior seen here numerically can be proved, optimal nonuniqueness examples for the Cauchy problem can be established, and two different solutions can exists for the same initial datum which is divergence-free, smooth away from the origin, compactly supported, and locally (−1)-homogeneous near the origin. In particular, assuming our (finite-dimensional) numerics represents faithfully the behavior of the full (infinitedimensional) system, the problem of uniqueness of the Leray–Hopf solutions (with non-smooth initial data) has a negative answer and, in addition, the perturbative arguments such those by Kato (Math Z 187(4):471–480, 1984, https://doi.org/ 10.1007/BF01174182) and Koch and Tataru (Adv Math 157(1):22–35, 2001, https://doi.org/10.1006/aima.2000.1937), or the weak-strong uniqueness results by Leray, Prodi, Serrin, Ladyzhenskaya and others, already give essentially optimal results. There are no singularities involved in the numerics, as we work only with smooth profile functions. It is conceivable that our calculations could be upgraded to a computer-assisted proof, although this would involve a substantial amount of additional work and calculations, including a much more detailed analysis of the asymptotic expansions of the solutions at large distances. 

We consider the equation q_1+qq_x=q_{xx} for quaternionic-valued functions and show that while singularities can develop from smooth compactly supported data, such situations are non-generic. The singularities will disappear under an arbitrary small “generic” smooth perturbation of the initial data. The equation is studied both on the real line and on the circle.