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We introduce an active vector system, which generalises both the 3D Euler equations and the electron–magnetohydrodynamic equations (E–MHD). We may as well view the system as singularised systems for the 3D Euler equations, in which case the equations of (E–MHD) correspond to the order two more singular one than the 3D Euler equations. The generalised surface quasi-geostrophic equation (gSQG) can be also embedded into a special case of our system when the unknown functions are constant in one coordinate direction. We investigate some basic properties of this system as well as the conservation laws. In the case when the system corresponds up to order one more singular than the 3D Euler equations, we prove local well-posedness in the standard Sobolev spaces. The proof crucially depends on a sharp commutator estimate similar to the one used for (gSQG) in Chae et al (2012 Commun. Pure Appl. Math. 65 1037–66). Since the system covers many areas of both physically and mathematically interesting cases, one can expect that there are various related problems to be investigated, parts of which are discussed here.

The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally m m -fold symmetric vortex patches with corners emanating from the origin are globally well-posed 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 global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness 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 ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar 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 long-time 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.

We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible $C^\infty([0,T)\times \mathbb{T}^d)\cap L^1([0,T]; C^{1-}(\mathbb{T}^d))$ velocity field which explicitly exhibits anomalous dissipation. As a consequence, this example also shows non-uniqueness of solutions to the transport equation with an incompressible $L^1([0,T]; C^{1-}(\mathbb{T}^d))$ drift, which is smooth except at one point in time. We also provide three sufficient conditions for anomalous dissipation provided solutions to the inviscid equation become singular in a controlled way. Finally, we discuss connections to the Obukhov-Corrsin monofractal theory of scalar turbulence along with other potential applications., Comment: It was pointed out to us by E. Bru\`{e} and Q-H. Nguyen that Conjecture 1.7, as stated, was false. The new version contains a modification of this conjecture which emerged after discussions with them

We consider the 3D incompressible Navier–Stokes equations under the following 2 + 1 2 2+\frac {1}{2} -dimensional situation: vertical vortex blob (quasi-streamwise vortices) being stretched by two-dimensional shear flow. We prove enhanced dissipation induced by such quasi-streamwise vortices.

In this paper, we investigate the initial value problem for the Euler–Riesz system, where the interaction forcing is given by ∇ ( − Δ ) s ρ for some − 1 s 0 , with s = − 1 corresponding to the classical Euler–Poisson system. We develop a functional framework to establish local-in-time existence and uniqueness of classical solutions for the Euler–Riesz system. In this framework, the fluid density could decay fast at infinity, and the Euler–Poisson system can be covered as a special case. Moreover, we prove local well-posedness for the pressureless Euler–Riesz system when the potential is repulsive, by observing hyperbolic nature of the system. Finally, we present sufficient conditions on the finite-time blowup of classical solutions for the isentropic/isothermal Euler–Riesz system with either attractive or repulsive interaction forces. The proof, which is based on estimates of several physical quantities, establishes finite-time blowup for a large class of initial data; in particular, it is not required that the density is of compact support.

The $m$-waves of Kelvin are uniformly rotating patch solutions of the 2D Euler equations with $m$-fold rotational symmetry for $m\geq 2$. For Kelvin waves sufficiently close to the disc, we prove a nonlinear stability result up to an arbitrarily long time in the $L^1$ norm of the vorticity, for $m$-fold symmetric perturbations. To obtain this result, we first prove that the Kelvin wave is a strict local maximizer of the energy functional in some admissible class of patches, which had been claimed by Wan in 1986. This gives an orbital stability result with a support condition on the evolution of perturbations, but using a Lagrangian bootstrap argument which traces the particle trajectories of the perturbation, we are able to drop the condition on the evolution. Based on this unconditional stability result, we establish that long time filamentation, or formation of long arms, occurs near the Kelvin waves, which have been observed in various numerical simulations. Additionally, we discuss stability of annular patches in the same variational framework., 37 pages, 2 figures, presentation updated, typos corrected

We study the evolution of corner-like patch solutions to the generalized SQG equations. Depending on the angle size and order of the velocity kernel, the corner instantaneously bents either downward or upward. In particular, we obtain the existence of strictly convex and smooth patch solutions which become immediately non-convex., 10 pages, 3 figures

We consider the incompressible Euler equations in the half cylinder $ \mathbb{R}_{>0}\times\mathbb{T}$. In this domain, any vorticity which is independent of $x_2$ defines a stationary solution. We prove that such a stationary solution is nonlinearly stable in a weighted $L^{1}$ norm involving the horizontal impulse, if the vorticity is non-negative and non-increasing in $x_1$. This includes stability of cylindrical patches $\{x_{1}0$. The stability result is based on the fact that such a profile is the unique minimizer of the horizontal impulse among all functions with the same distribution function. Based on stability, we prove existence of vortex patches in the half cylinder that exhibit infinite perimeter growth in infinite time., Comment: 17 pages, 2 figures, changed title to specify our main idea