We show that certain singular structures (Hölderian cusps and mild divergences) are transported by the flow of homeomorphisms generated by an Osgood velocity field. The structure of these singularities is related to the modulus of continuity of the velocity and the results are shown to be sharp in the sense that slightly more singular structures cannot generally be propagated. For the 2D Euler equation, we prove that certain singular structures are preserved by the motion, e.g. a system of
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Abstract vortices (and those that are slightly less singular) travel with the fluid in a nonlinear fashion, up to bounded perturbations. We also give stability results for weak Euler solutions away from their singular set.$$\log \log _+(1/x)$$ $log{log}_{+}(1/x\left\right)$ 
Abstract We study the behavior of solutions to the incompressible 2
d Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier–Stokes problems. We exhibit a large family of new, nontrivial stationary states that are arbitrarily close to the Kolmogorov flow on the square torus in analytic regularity. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: there the linearized problem exhibits an “inviscid damping” mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. Our results show that such a simple description of the longtime behavior is not possible for solutions near the Kolmogorov flow on$$\mathbb {T}^2$$ ${T}^{2}$ . Our construction of the new stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on$$\mathbb {T}^2$$ ${T}^{2}$ , and we also show a lack of correspondence between the linearized description of the set of steady states and its true nonlinear structure. Both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different. We show that the only stationary states near them must indeed be shears, even in relatively low regularity. In addition, we show that this behavior is mirrored closely in the related Navier–Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on$$\mathbb {T}^2$$ ${T}^{2}$ .$${\mathbb T}^2$$ ${T}^{2}$ 
The purpose of this work is to discuss the wellposedness theory of singular vortex patches. Our main results are of two types: wellposedness and illposedness. On the wellposedness side, we show that globally m m fold symmetric vortex patches with corners emanating from the origin are globally wellposed in natural regularity classes as long as m ≥ 3. m\geq 3. In this case, all of the angles involved solve a closed ODE system which dictates the globalintime dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global wellposedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the illposedness side, we show that any other type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angle π 2 \frac {\pi }{2} for all time. Even in the case of vortex patches with corners of angle π 2 \frac {\pi }{2} or with corners which are only locally m m fold symmetric, we prove that they are generically illposed. We expect that in these cases of illposedness, the vortex patches actually cusp immediately in a selfsimilar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches, we discuss the longtime behavior of symmetric vortex patches with corners and use them to construct patches on R 2 \mathbb {R}^2 with interesting dynamical behavior such as cusping and spiral formation in infinite time.more » « less

We consider transport of a passive scalar advected by an irregular divergencefree vector field. Given any nonconstant initial data ρ ¯ ∈ H loc 1 ( R d ) , d ≥ 2 , we construct a divergencefree advecting velocity field v (depending on ρ ¯ ) for which the unique weak solution to the transport equation does not belong to H loc 1 ( R d ) for any positive time. The velocity field v is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space W s , p that does not embed into the Lipschitz class. The velocity field v is constructed by pulling back and rescaling a sequence of sine/cosine shear flows on the torus that depends on the initial data. This loss of regularity result complements that in Ann. PDE , 5(1):Paper No. 9, 19, 2019. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.more » « less