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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

Vortex thinning is one of the main mechanisms of two-dimensional turbulence. By direct numerical simulation to the two-dimensional Navier–Stokes equations with small-scale forcing and large-scale damping, Xiao et al (2009 J. Fluid Mech. 619 1–44) found an evidence that inverse energy cascade may proceed with the vortex thinning mechanism. On the other hand, Alexakis and Doering (2006 Phys. Lett. A 359 652–657) calculated the upper bound of the bulk averaged enstrophy dissipation rate of the steady-state two dimensional turbulence. In this paper, we show that vortex thinning induces enhanced dissipation with strictly slower vanishing order of the enstrophy dissipation than Re−1.

The generalized Constantin–Lax–Majda (gCLM) equation was introduced to model the competing effects of advection and vortex stretching in hydrodynamics. Recent investigations revealed possible connections with the two-dimensional turbulence. With this connection in mind, we consider the steady problem for the viscous gCLM equations on T : a v ω x − v x ω = ν Δ ω + f , v = ( − Δ ) − 1 2 ω , where a ∈ R is the parameter measuring the relative strength between advection and stretching, ν > 0 is the viscosity constant, and f is a given O(1)-forcing independent of ν. For some range of parameters, we establish existence and uniqueness of stationary solutions. We then numerically investigate the behaviour of solutions in the vanishing viscosity limit, where bifurcations appear, and new solutions emerge. When the parameter a is away from [−1/2, 1], we verify that there is convergence towards smooth stationary solutions for the corresponding inviscid equation. Moreover, we analyse the inviscid limit in the fractionally dissipative case, as well as the behaviour of singular limiting solutions.

In a companion paper, we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in the companion work. In particular, we give examples of time-periodic behavior, cusp formation in infinite time at an exponential rate, and spiral formation in infinite time., Comment: 17 pages, 6 figures

We consider the 3D incompressible Navier–Stokes equations under the following $$2+\frac{1}{2}$$ -dimensional situation: small-scale horizontal vortex blob being stretched by large-scale, anti-parallel pairs of vertical vortex tubes. We prove enhanced dissipation induced by such vortex-stretching.

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

We perform asymptotic analysis for the Euler–Riesz system posed in either $${\mathbb {T}}^d$$ or $${\mathbb {R}}^d$$ in the high-force regime and establish a quantified relaxation limit result from the Euler–Riesz system to the fractional porous medium equation. We provide a unified approach for asymptotic analysis regardless of the presence of pressure in the case of repulsive Riesz interactions, based on the modulated energy estimates, the Wasserstein distance of order 2, and the bounded Lipschitz distance. For the attractive Riesz interaction case, we consider the periodic domain and estimate a lower bound on the modulated internal energy to handle the modulated interaction energy.

In this note, we construct solutions to the 2D Euler equations which belong to the Yudovich class but lose $W^{1,p}$ regularity continuously with time., 12 pages, 1 figure, to appear in J. Math. Fluid Mech

We prove finite-time singularity formation for De Gregorio's model of the three-dimensional vorticity equation in the class of $L^p\cap C^\alpha(\mathbb{R})$ vorticities for some $\alpha>0$ and $p0$., Comment: 48 pages

We show existence of self-similar solutions satisfying Kolmogorov's scaling for generalized dyadic models of the Euler equations, extending a result of Barbato, Flandoli, and Morandin. The proof is based on the analysis of certain dynamical systems on the plane., Comment: 10 pages, 1 figure

We consider the Keller-Segel system of consumption type coupled with an incompressible fluid equation. The system describes the dynamics of oxygen and bacteria densities evolving within a fluid. We establish local well-posedness of the system in Sobolev spaces for partially inviscid and fully inviscid cases. In the latter, additional assumptions on the initial data are required when either the oxygen or bacteria density touches zero. Even though the oxygen density satisfies a maximum principle due to consumption, we prove finite time blow-up of its $C^{2}$--norm with certain initial data., 39 pages, Comm. Math Phys., to appear

We consider the fractional porous medium flow introduced by Caffarelli-Vazquez and obtain local in time existence, uniqueness, and blow-up criterion for smooth solutions. The proof is based on establishing a commutator estimate involving fractional Laplacian operators., 13 pages

We prove linear in time filamentation for perturbations of the Lamb dipole, which is a traveling wave solution to the incompressible Euler equations in $\mathbb{R}^2$. The main ingredient is a recent nonlinear orbital stability result by Abe-Choi. As a consequence, we obtain linear in time growth for the vorticity gradient for all times, for certain smooth and compactly supported initial vorticity in $\mathbb{R}^2$. The construction carries over to some generalized SQG equations., Comment: 24 pages

The global regularity problem for the Boussinesq system is a well known open problem in mathematical fluid dynamics. As a follow up to our work \cite{EJSI}, we give examples of finite-energy and Lipschitz continuous velocity field and density $(u_0,\rho_0)$ which are $C^\infty$-smooth away from the origin and belong to a natural local well-posedness class for the Boussinesq equation whose corresponding local solution becomes singular in finite time. That is, while the sup norm of the gradient of the velocity field and the density remain finite on the time interval $t\in [0,1)$, both quantities become infinite as $t\rightarrow 1$. The key is to use scale-invariant solutions similar to those introduced in \cite{EJSI}. The proof consists of three parts: local well-posedness for the Boussinesq equation in critical spaces, the analysis of certain special infinite-energy solutions belonging to those critical spaces, and finally a cut-off argument to ensure finiteness of energy. All of this is done on spatial domains $\{(x_1,x_2): x_1 \ge \gamma|x_2|\}$ for any $\gamma > 0$ so that we can get arbitrarily close to the half-space case. We remark that the $2D$ Euler equation is globally well-posed in all of the situations we look at, so that the singularity is not coming from the domain or the lack of smoothness on the data but from the vorticity amplification due to the presence of a density gradient. It is conceivable that our methods can be adapted to produce finite-energy $C^\infty$ solutions on $\mathbb{R}^2_+$ which become singular in finite time., Comment: 40 pages, 1 figure

We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: $\{ (x_1,x_2,x_3): 0, Comment: 53 pages, 5 figures

For all $\epsilon>0$, we prove the existence of finite-energy strong solutions to the axi-symmetric $3D$ Euler equations on the domains $ \{(x,y,z)\in\mathbb{R}^3: (1+\epsilon|z|)^2\leq x^2+y^2\}$ which become singular in finite time. We further show that solutions with 0 swirl are necessarily globally regular. The proof of singularity formation relies on the use of approximate solutions at exactly the critical regularity level which satisfy a $1D$ system which has solutions which blow-up in finite time. The construction bears similarity to our previous result on the Boussinesq system \cite{EJB} though a number of modifications must be made due to anisotropy and since our domains are not scale-invariant. This seems to be the first construction of singularity formation for finite-energy strong solutions to the actual $3D$ Euler system., Comment: 46 pages, 1 figure. arXiv admin note: text overlap with arXiv:1708.09372

We consider the 2D Euler equation with periodic boundary conditions in a family of Banach spaces based on the Fourier coefficients, and show that it is ill-posed in the sense that 'norm inflation' occurs. The proof is based on the observation that the evolution of certain perturbations of the 'Kolmogorov flow' given in velocity by \begin{document}$U(x,y) = \left( {\begin{array}{*{20}{c}}{\cos \;y}\\0\end{array}} \right)$ \end{document} can be well approximated by the linear Schrodinger equation, at least for a short period of time.

In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness. Perhaps the most striking conclusion of this article is that the Cauchy problems for the Hall-MHD (either viscous or inviscid) and the electron-MHD equations, under one translational symmetry, are ill-posed near the trivial solution in any sufficiently high regularity Sobolev space and even in Gevrey spaces. This result holds despite obvious wellposedness of the linearized equations near the trivial solution, as well as conservation of the nonlinear energy, by which the $L^{2}$ norm (energy) of the solution stays constant in time. The core illposedness mechanism is degeneration of certain high frequency wave packet solutions to the linearization around a class of linearly degenerate stationary solutions of these equations, which are essentially dispersive equations with degenerate principal symbols. The method developed in this work is sharp and robust, in that we also prove nonlinear $H^{s}$-illposedness in the presence of fractional dissipation of any order less than 1, matching known wellposedness results. The results in this article are complemented by a companion work, where we provide geometric conditions on the initial magnetic field that ensure wellposedness(!) of the Cauchy problem. In particular, in stark contrast to the results here, it is shown in the companion work that the nonlinear Cauchy problems are well-posed near any nonzero constant magnetic field., Comment: 80 pages, 1 figure

We partially answer a question raised by Kiselev and Zlatos in \cite{MR2180809}; in the generalized dyadic model of the Euler equation, a blow-up of $H^{1/3+\delta}$-norm occurs. We recover a few previous blow-up results for various related dyadic models as corollaries., Comment: 8 pages

For the 2D Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to $$L^\infty \cap H^1$$ but escapes $$H^1$$ immediately for $$t>0$$ . Our main observation is that a localized chunk of vorticity bounded in $$L^\infty \cap H^1$$ with odd-odd symmetry is able to generate a hyperbolic flow with large velocity gradient at least for a short period of time, which stretches the vorticity gradient.