Partial regularity of Leray-Hopf weak solutions to the incompressible Navier-Stokes equations with hyperdissipation

We show that if $u$ is a Leray-Hopf weak solution to the incompressible Navier--Stokes equations with hyperdissipation $\alpha \in (1,5/4)$ then there exists a set $S\subset \mathbb{R}^3$ such that $u$ remains bounded outside of $S$ at each blow-up time, the Hausdorff dimension of $S$ is bounded above by $5-4\alpha$ and its box-counting dimension is bounded by $(-16\alpha^2 + 16\alpha +5)/3$. Our approach is inspired by the ideas of Katz & Pavlovi\'c (Geom. Funct. Anal., 2002).Comment: 37 pages, 3 figure

A sufficient integral condition for local regularity of solutions to the surface growth model

The surface growth model, $u_t + u_{xxxx} + \partial_{xx} u_x^2 =0$, is a one-dimensional fourth order equation, which shares a number of striking similarities with the three-dimensional incompressible Navier--Stokes equations, including the results regarding existence and uniqueness of solutions and the partial regularity theory. Here we show that a weak solution of this equation is smooth on a space-time cylinder $Q$ if the Serrin condition $u_x\in L^{q'}L^q (Q)$ is satisfied, where $q,q'\in [1,\infty ]$ are such that either $1/q+4/q'<1$ or $1/q+4/q'=1$, $q'<\infty$.Comment: 18 page

Weak solutions to the Navier-Stokes inequality with arbitrary energy profiles

In a recent paper, Buckmaster & Vicol (arXiv:1709.10033) used the method of convex integration to construct weak solutions $u$ to the 3D incompressible Navier-Stokes equations such that $\| u(t) \|_{L^2} =e(t)$ for a given non-negative and smooth energy profile $e: [0,T]\to \mathbb{R}$. However, it is not known whether it is possible to extend this method to construct nonunique suitable weak solutions (that is weak solutions satisfying the strong energy inequality (SEI) and the local energy inequality (LEI)), Leray-Hopf weak solutions (that is weak solutions satisfying the SEI), or at least to exclude energy profiles that are not nonincreasing. In this paper we are concerned with weak solutions to the Navier-Stokes inequality on $\mathbb{R}^3$, that is vector fields that satisfy both the SEI and the LEI (but not necessarily solve the Navier-Stokes equations). Given $T>0$ and a nonincreasing energy profile $e\colon [0,T] \to [0,\infty )$ we construct weak solution to the Navier-Stokes inequality that are localised in space and whose energy profile $\| u(t)\|_{L^2 (\mathbb{R}^3 )}$ stays arbitrarily close to $e(t)$ for all $t\in [0,T]$. Our method applies only to nonincreasing energy profiles. The relevance of such solutions is that, despite not satisfying the Navier-Stokes equations, they satisfy the partial regularity theory of Caffarelli, Kohn & Nirenberg (Comm. Pure Appl. Math., 1982). In fact, Scheffer's constructions of weak solutions to the Navier-Stokes inequality with blow-ups (Comm. Math. Phys., 1985 & 1987) show that the Caffarelli, Kohn & Nirenberg's theory is sharp for such solutions. Our approach gives an indication of a number of ideas used by Scheffer. Moreover, it can be used to obtain a stronger result than Scheffer's. Namely, we obtain weak solutions to the Navier-Stokes inequality with both blow-up and a prescribed energy profile.Comment: 26 pages, 4 figures. arXiv admin note: text overlap with arXiv:1709.0060

An improvement to Prandtl's 1933 model for minimizing induced drag

We consider Prandtl's 1933 model for calculating circulation distribution function $\Gamma$ of a finite wing which minimizes induced drag, under the constraints of prescribed total lift and moment of inertia. We prove existence of a global minimizer of the problem without the restriction of nonnegativity $\Gamma\geq 0$ in an appropriate function space. We also consider an improved model, where the prescribed moment of inertia takes into account the bending moment due to the weight of the wing itself, which leads to a more efficient solution than Prandtl's 1933 result.Comment: 10 pages, 2 figure

On weak solutions to the Navier-Stokes inequality with internal singularities

We construct weak solutions to the Navier-Stokes inequality, $$u\cdot \left(\partial_t u -\nu \Delta u + (u\cdot \nabla) u +\nabla p \right) \leq 0$$ in $\mathbb{R}^3$, which blow up at a single point $(x_0,T_0)$ or on a set $S \times \{T_0 \}$, where $S\subset \mathbb{R}^3$ is a Cantor set whose Hausdorff dimension is at least $\xi$ for any preassigned $\xi\in (0,1)$. Such solutions were constructed by Scheffer, Comm. Math. Phys., 1985 & 1987. Here we offer a simpler perspective on these constructions. We sharpen the approach to construct smooth solutions to the Navier-Stokes inequality on the time interval $[0,1]$ satisfying the "approximate equality" $$\left\| u\cdot \left(\partial_t u-\nu \Delta u + (u\cdot \nabla) u +\nabla p \right) \right\|_{L^\infty}\leq \vartheta,$$ and the "norm inflation" $\| u(1) \|_{L^\infty} \geq \mathcal{N} \| u(0) \|_{L^\infty}$ for any preassigned $\mathcal{N}>0$, $\vartheta >0$. Furthermore we extend the approach to construct a weak solution to the Euler inequality $$u\cdot \left(\partial_t u+ (u\cdot \nabla) u +\nabla p \right) \leq 0,$$ which satisfies the approximate equality with $\nu =0$ and blows up on the Cantor set $S\times \{T_0 \}$ as above.Comment: 86 pages, 23 figure

Quantitative transfer of regularity of the incompressible Navier-Stokes equations from R3\mathbb{R}^3 to the case of a bounded domain

Let $u_0\in C_0^5 ( B_{R_0})$ be divergence-free and suppose that $u$ is a strong solution of the three-dimensional incompressible Navier-Stokes equations on $[0,T]$ in the whole space $\mathbb{R}^3$ such that $\| u \|_{L^\infty ((0,T);H^5 (\mathbb{R}^3 ))} + \| u \|_{L^\infty ((0,T);W^{5,\infty }(\mathbb{R}^3 ))} \leq M <\infty$. We show that then there exists a unique strong solution $w$ to the problem posed on $B_R$ with the homogeneous Dirichlet boundary conditions, with the same initial data and on the same time interval for $R\geq \max(1+R_0, C(a) C(M)^{1/a} \exp ({CM^4T/a})) )$ for any $a\in [0,3/2)$, and we give quantitative estimates on $u-w$ and the corresponding pressure functions.Comment: 16 page

Leray's fundamental work on the Navier-Stokes equations: a modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace"

This article offers a modern perspective which exposes the many contributions of Leray in his celebrated work on the Navier--Stokes equations from 1934. Although the importance of his work is widely acknowledged, the precise contents of his paper are perhaps less well known. The purpose of this article is to fill this gap. We follow Leray's results in detail: we prove local existence of strong solutions starting from divergence-free initial data that is either smooth, or belongs to $H^1$, $L^2\cap L^p$ (with $p\in(3,\infty]$), as well as lower bounds on the norms $\| \nabla u (t) \|_2$, $\| u(t) \|_p$ ($p\in(3,\infty]$) as $t$ approaches a putative blow-up time. We show global existence of a weak solution and weak-strong uniqueness. We present Leray's characterisation of the set of singular times for the weak solution, from which we deduce that its upper box-counting dimension is at most $\tfrac{1}{2}$. Throughout the text we provide additional details and clarifications for the modern reader and we expand on all ideas left implicit in the original work, some of which we have not found in the literature. We use some modern mathematical tools to bypass some technical details in Leray's work, and thus expose the elegance of his approach.Comment: 81 pages. All comments are welcom

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

A partial uniqueness result and an asymptotically sharp nonuniqueness result for the Zhikov problem on the torus

We consider the stationary diffusion equation $-\mathrm{div} (\nabla u + bu )=f$ in $n$-dimensional torus $\mathbb{T}^n$, where $f\in H^{-1}$ is a given forcing and $b\in L^p$ is a divergence-free drift. Zhikov (Funkts. Anal. Prilozhen., 2004) considered this equation in the case of a bounded, Lipschitz domain $\Omega \subset \mathbb{R}^n$, and proved existence of solutions for $b\in L^{2n/(n+2)}$, uniqueness for $b\in L^2$, and has provided a point-singularity counterexample that shows nonuniqueness for $b\in L^{3/2-}$ and $n=3,4,5$. We apply a duality method and a DiPerna-Lions-type estimate to show uniqueness of the solutions constructed by Zhikov for $b\in W^{1,1}$. We use a Nash iteration to demonstrate sharpness of this result, and also show that solutions in $H^1\cap L^{p/(p-1)}$ are flexible for $b\in L^p$, $p\in [1,2(n-1)/(n+1))$; namely we show that the set of $b\in L^p$ for which nonuniqueness in the class $H^1\cap L^{p/(p-1)}$ occurs is dense in the divergence-free subspace of $L^p$.Comment: 16 page

Weak solutions to the Navier-Stokes inequality with arbitrary energy profiles

In a recent paper, Buckmaster and Vicol (Ann Math (2) 189(1):101–144, 2019) used the method of convex integration to construct weak solutions u to the 3D incompressible Navier–Stokes equations such that ∥u(t)∥L2=e(t) for a given non-negative and smooth energy profile e:[0,T]→R . However, it is not known whether it is possible to extend this method to construct nonunique suitable weak solutions (that is weak solutions satisfying the strong energy inequality (SEI) and the local energy inequality (LEI)), Leray–Hopf weak solutions (that is weak solutions satisfying the SEI), or at least to exclude energy profiles that are not nonincreasing. In this paper we are concerned with weak solutions to the Navier–Stokes inequality on R3 , that is vector fields that satisfy both the SEI and the LEI (but not necessarily solve the Navier–Stokes equations). Given T>0 and a nonincreasing energy profile e:[0,T]→[0,∞) we construct weak solution to the Navier–Stokes inequality that are localised in space and whose energy profile ∥u(t)∥L2(R3) stays arbitrarily close to e(t) for all t∈[0,T] . Our method applies only to nonincreasing energy profiles. The relevance of such solutions is that, despite not satisfying the Navier–Stokes equations, they satisfy the partial regularity theory of Caffarelli et al. (Commun Pure Appl Math 35(6):771–831, 1982). In fact, Scheffer’s constructions of weak solutions to the Navier–Stokes inequality with blow-ups (Commun Math Phys 101(1):47–85, 1985; Commun Math Phys 110(4): 525–551, 1987) show that the Caffarelli, Kohn & Nirenberg’s theory is sharp for such solutions. Our approach gives an indication of a number of ideas used by Scheffer. Moreover, it can be used to obtain a stronger result than Scheffer’s. Namely, we obtain weak solutions to the Navier–Stokes inequality with both blow-up and a prescribed energy profile