Search

We introduce a notion of stability for non-autonomous Hamiltonian flows on two-dimensional annular surfaces. This notion of stability is designed to capture the sustained twisting of particle trajectories. The main Theorem is applied to establish a number of results that reveal a form of irreversibility in the Euler equations governing the motion of an incompressible and inviscid fluid. In particular, we show that nearby general stable steady states (i) all fluid flows exhibit indefinite twisting (ii) vorticity generically exhibits gradient growth and wandering. We also give examples of infinite time gradient growth for smooth solutions to the SQG equation and of smooth vortex patches that entangle and develop unbounded perimeter in infinite time. [ABSTRACT FROM AUTHOR]

We prove the global existence and uniqueness of solutions to the Vlasov–Riesz–Fokker–Planck system around the global Maxwellian in the periodic spatial domain. Depending on the order of Riesz potential, we present two frameworks for the construction of global-in-time solutions with Sobolev and analytic regularity. The analytic function framework covers the Vlasov–Dirac–Benney–Fokker–Planck system. Furthermore, we show the exponential decay of solutions toward the global Maxwellian. Our result is generalized to the whole space case in which the decay rate of convergence is algebraic. [ABSTRACT FROM AUTHOR]

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 bends either downward or upward. In particular, we obtain a new proof of the existence of strictly convex and smooth patch solutions which become immediately non-convex. [ABSTRACT FROM AUTHOR]

We consider the incompressible Euler equations in the half cylinder R > 0 × 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 nonnegative and non-increasing in x 1 . This includes stability of cylindrical patches { x 1 < α } , α > 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. [ABSTRACT FROM AUTHOR]

The m-waves of Kelvin are uniformly rotating patch solutions of the 2D Euler equations with m-fold rotational symmetry for m ≥ 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible C ∞ ([ 0 , T) × T d) ∩ L 1 ([ 0 , T ] ; C 1 - (T d)) velocity field which explicitly exhibits anomalous dissipation. As a consequence, this example also shows the non-uniqueness of solutions to the transport equation with an incompressible L 1 ([ 0 , T ] ; C 1 - (T d)) drift, which is smooth except at one point in time. We also give a sufficient condition for anomalous dissipation based on solutions to the inviscid equation becoming singular in a controlled way. Finally, we discuss connections to the Obukhov-Corrsin monofractal theory of scalar turbulence along with other potential applications. [ABSTRACT FROM AUTHOR]

We perform asymptotic analysis for the Euler–Riesz system posed in either T d or 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the 3D incompressible Navier–Stokes equations under the following 2 + 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. [ABSTRACT FROM AUTHOR]

We prove finite-time singularity formation for De Gregorio's model of the three-dimensional vorticity equation in the class of L p ∩ C α (R) vorticities for some α > 0 and p < ∞ . We also prove finite-time singularity formation from smooth initial data for the Okamoto–Sakajo–Wunsch models in a new range of parameter values. As a consequence, we have finite-time singularity for certain infinite-energy solutions of the surface quasi-geostrophic equation which are C α -regular. One of the difficulties in the models we consider is that there are competing nonlocal stabilizing effects (advection) and destabilizing effects (vortex stretching) which are of the same size in terms of scaling. Hence, it is difficult to establish the domination of one effect over the other without having strong control of the solution. We conjecture that strong solutions to the De Gregorio model exhibit the following behavior: for each 0 < α < 1 there exists an initial ω 0 ∈ C α (R) which is compactly supported for which the solution becomes singular in finite-time; on the other hand, solutions to De Gregorio's equation are global whenever ω 0 ∈ L p ∩ C 1 (R) for some p < ∞ . Such a dichotomy seems to be a genuinely non-linear effect which cannot be explained merely by scaling considerations since C α spaces are scaling subcritical for each α > 0 . [ABSTRACT FROM AUTHOR]

We partially answer a question raised by Kiselev and Zlatos (Int Math Res Not 2005(38):2315-2339, ); 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. [ABSTRACT FROM AUTHOR]