In this paper we study the large distance asymptotics of small steady solutions of the 3d Navier Stokes equation in exterior domains. It was proved by Korolev and the second author that the leading term is given by the Landau solution, and it was conjectured that the next order term should be O(1/|x|^2) as x→∞. We confirm that this is indeed the case and we compute the next order asymptotics in terms of eigenvalues of a suitably constructed linearized operator around the Landau solution on the unit sphere. While the decay of some of the terms is precisely O(1/|x|^2), the the decay of other terms is slightly accelerated.

For the incompressible Navier-Stokes equations in $R^3$ with low viscosity $\nu>0$, we consider the Cauchy problem with initial vorticity $\omega_0$ that represents an infinitely thin vortex filament of arbitrary given strength $\Gamma$ supported on a circle. The vorticity field $\omega(x,t)$ of the solution is smooth at any positive time and corresponds to a vortex ring of thickness $\sqrt{\nu t}$ 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 $\omega(x,t)$ is well-approximated on a large time interval by $\omega_{lin}(x-a(t),t)$, where $\omega_{lin}(\cdot,t)=\exp(\nu t\Delta)\omega_0$ is the solution of the heat equation with initial data $\omega_0$, and $\dot a(t)$ 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 $\nu=0$ 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 $\nu > 0$, 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 the Navier-Stokes equation at low viscosity. In particular, we revisit the classical example of Oseen's vortex, providing a new stability proof with 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 consider the Cauchy problem for the incompressible Navier–Stokes equations in ℝ^3 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 Šverák (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 non-uniqueness 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 (infinite-dimensional) 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 $$q_t+qq_x=q_{xx}$$ for $$q:{{\mathbf {R}}}\times (0,\infty )\rightarrow {\mathbf {H}}$$ (the quaternions), 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. Similar results are also established for the same equation in $$\mathbf{S}^1\times (0,\infty )$$ , where $$\mathbf{S}^1$$ is the standard one-dimensional circle.

The incompressible Navier-Stokes equations in R^3 are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on uniqueness, as existence has already been established in [10]. The main difficulty which has to be overcome is that the nonlinear regime for such flows is outside of applicability of standard perturbation theory, even for short times. The solutions we consider are archetypal examples of viscous vortex rings, and can be thought of as axisymmetric analogues of the self-similar Lamb-Oseen vortices in two-dimensional flows. Our method provides the leading term in a fixed-viscosity short-time asymptotic expansion of the solution, and may in principle be extended so as to give a rigorous justification, in the axisymmetric situation, of higher-order formal asymptotic expansions that can be found in the literature [7].

We study a modification due to De Gregorio of the Constantin–Lax–Majda (CLM) model \({\omega_{t}}\) = \({\omega H\omega}\) on the unit circle. The De Gregorio equation is \({\omega_{t} +u\omega_{x}-{u}_{x}\omega = 0, {u}_{x} = H\omega }\). In contrast with the CLM model, numerical simulations suggest that the solutions of the De Gregorio model with smooth initial data exist globally for all time, and generically converge to equilibria when \({t \rightarrow \pm \infty}\), in a way resembling inviscid damping. We prove that such behavior takes place near a manifold of equilibria.

We discuss several model PDEs motivated by the incompressible Navier-Stokes equations. Some of the PDEs appear to be quite simpler, but basic questions about them are still open. In the last section we discuss uniqueness of weak solutions of the 3d incompressible Navier-Stokes in a natural energy class.

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We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n = 2 is still harder.

We consider stochastic perturbations of geodesic flow for left-invariant metrics on finite-dimensional Lie groups and study the Hörmander condition and some properties of the solutions of the corresponding Fokker–Planck equations.

We consider the Cauchy problem for the Navier-Stokes equation in (3)x]0,[ with the initial datum , a critical space containing nontrivial (-1)-homogeneous fields. For small one can get global well-posedness by perturbation theory. When is not small, the perturbation theory no longer applies and, very likely, the local-in-time well-posedness and uniqueness fails. One can still develop a good theory of weak solutions with the following stability property: If u((n)) are weak solutions corresponding the the initial datum , and converge weakly* in to u(0), then a suitable subsequence of u((n)) converges to a weak solution u corresponding to the initial condition u(0). This is of interest even in the special case u(0)0.

Collating different aspects of Vector-valued Partial Differential Equations and Applications, this volume is based on the 2013 CIME Course with the same name which took place at Cetraro, Italy, under

The aim of the note is to discuss different definitions of solutions to the Cauchy problem for the Navier–Stokes equations with the initial data belonging to the Lebesgue space L3(R3)

In the first part of the notes we discuss the problem of long-time behavior of some infinite-dimensional Hamiltonian system from the point of view of statistical mechanics. In the second part we discuss the issue of well-posedness of the Leray-Hopf weak solutions in the energy space in the context of recent developments concerning scale-invariant solutions.

In connection with the recent proposal for possible singularity formation at the boundary for solutions of three-dimensional axisymmetric incompressible Euler's equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite-time blowup from smooth data. (c) 2017 Wiley Periodicals, Inc.

We consider various questions about the 2d incompressible Navier-Stokes and Euler equations on a torus when dissipation is removed from or added to some of the Fourier modes.