Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary

We generalize two results in the Navier-Stokes regularity theory whose proofs rely on `zooming in' on a presumed singularity to the local setting near a curved portion $\Gamma \subset \partial\Omega$ of the boundary. Suppose that $u$ is a boundary suitable weak solution with singularity $z^* = (x^*,T^*)$, where $x^* \in \Omega \cup \Gamma$. Then, under weak background assumptions, the $L_3$ norm of $u$ tends to infinity in every ball centered at $x^*$: \begin{equation*} \lim_{t \to T^*_-} \lVert u(\cdot, t)\rVert_{L_{3}\left(\Omega \cap B(x^*,r)\right)} = \infty \quad \forall r > 0. \end{equation*} Additionally, $u$ generates a non-trivial `mild bounded ancient solution' in $\mathbb{R}^3$ or $\mathbb{R}^3_+$ through a rescaling procedure that `zooms in' on the singularity. Our proofs rely on a truncation procedure for boundary suitable weak solutions. The former result is based on energy estimates for $L_3$ initial data and a Liouville theorem. For the latter result, we apply perturbation theory for $L_\infty$ initial data based on linear estimates due to K. Abe and Y. Giga

Non-uniqueness of Leray solutions to the hypodissipative Navier-Stokes equations in two dimensions

We exhibit non-unique Leray solutions of the forced Navier-Stokes equations with hypodissipation in two dimensions. Unlike the solutions constructed in \cite{albritton2021non}, the solutions we construct live at a supercritical scaling, in which the hypodissipation formally becomes negligible as $t \to 0^+$. In this scaling, it is possible to perturb the Euler non-uniqueness scenario of Vishik \cite{Vishik1,Vishik2} to the hypodissipative setting at the nonlinear level. Our perturbation argument is quasilinear in spirit and circumvents the spectral theoretic approach to incorporating the dissipation in \cite{albritton2021non}.Comment: 17 page

Linear and nonlinear instability of vortex columns

We consider vortex column solutions to the $3$D Euler equations, that is, $v = V(r) e_\theta + W(r) e_z$. Under sufficient conditions on $V$ and $W$, we rigorously construct a countable family of unstable modes with $O(1)$ growth rate which concentrate on a ring $r= r_0$ asymptotically as the azimuthal and axial wavenumbers $n, \alpha \to \infty$ with a fixed ratio. These `ring modes' were predicted by Liebovich and Stewartson (\emph{J. Fluid Mech.} 126, 1983) by formal asymptotic analysis. We construct them with a gluing procedure and the Lyapunov-Schmidt reduction. Finally, we prove that the existence of an unstable mode implies nonlinear instability for vortex columns. %For quasilinear equations, this is not obvious. In particular, our analysis yields nonlinear instability for the \emph{trailing vortex}, that is, $V(r) := \frac{q}{r} (1-\mathrm{e}^{-r^2})$ and $W(r) := \mathrm{e}^{-r^2}$, where $0 < q \ll 1$.Comment: 35 pages, 2 figure

Gluing non-unique Navier-Stokes solutions

We construct non-unique Leray solutions of the forced Navier-Stokes equations in bounded domains via gluing methods. This demonstrates a certain locality and robustness of the non-uniqueness discovered by the authors in [1]

Regularity aspects of the Navier-Stokes equations in critical spaces

University of Minnesota Ph.D. dissertation. August 2020. Major: Mathematics. Advisor: Vladimir Sverak. 1 computer file (PDF); ii, 158 pages.For better or for worse, our current understanding of the Navier-Stokes regularity problem is intimately connected with certain dimensionless quantities known as critical norms. In this thesis, we concern ourselves with one of the most basic questions about Navier-Stokes regularity: How must the critical norms behave at a potential Navier-Stokes singularity? In Chapter 2, we give a broad overview of the Navier-Stokes theory necessary to answer this question. This chapter is suitable for newcomers to the field. Next, we present two of our published papers [4,5] which answer this question in the context of homogeneous Besov spaces. In Chapter 3, we demonstrate that the critical Besov norms $\| u(\cdot,t) \|_{\dot B^{-1+3/p}_{p,q}(\R^3)}$, $p,q \in (3,+\infty)$, must tend to infinity at a potential singularity. Our proof has been streamlined from the published version [4]. In Chapter 4 (joint work with Tobias Barker), we develop a framework of global weak Besov solutions with initial data belonging to $\dot B^{-1+3/p}_{p,\infty}(\R^3)$, $p \in (3,+\infty)$. To illustrate this framework, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. This chapter has been reproduced from the published version [5]

On the stabilizing effect of swimming in an active suspension

We consider a kinetic model of an active suspension of rod-like microswimmers. In certain regimes, swimming has a stabilizing effect on the suspension. We quantify this effect near homogeneous isotropic equilibria $\overline{\psi} = \text{const}$. Notably, in the absence of particle (translational and orientational) diffusion, swimming is the only stabilizing mechanism. On the torus, in the non-diffusive regime, we demonstrate linear Landau damping up to the stability threshold predicted in the applied literature. With small diffusion, we demonstrate nonlinear stability of arbitrary equilibrium values for pullers (rear-actuated swimmers) and enhanced dissipation for both pullers and pushers (front-actuated swimmers) at small concentrations. On the whole space, we prove nonlinear stability of the vacuum equilibrium due to generalized Taylor dispersion.Comment: 39 pages, no figure