Active Scalar Equations Research Articles

On the Regularity of Solutions to Some Active Scalar Equations on Bounded Domains

On the Regularity of Solutions to Some Active Scalar Equations on Bounded Domains

A generalized Biot–Savart law and its application to the active scalar equations

A generalized Biot–Savart law and its application to the active scalar equations

Non-uniqueness of forced active scalar equations with even drift operators

We consider forced active scalar equations with even and homogeneous degree 0 drift operator on Td. Inspired by the non-uniqueness construction for dyadic fluid models (Dai and Friedlander, 2022; Filonov and Khodunov, 2021), by implementing a sum-difference convex integration scheme we obtain non-unique weak solutions for the active scalar equation in space Ct0Cxα with α<12d+1. Without external forcing, Isett and Vicol (2015) constructed non-unique weak solutions for such active scalar equations with spatial regularity Cxα for α<14d+1.

Trend to Equilibrium for Flows With Random Diffusion

Abstract Motivated by the possibility of noise to cure equations of finite-time blowup, the recent work [ 90] by the second and third named authors showed that with quantifiable high probability, random diffusion restores global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as the Patlak–Keller–Segel equation. A question left open is the asymptotic behavior of the solutions, in particular, whether they converge to a steady state. We answer this question by showing that the solutions from [ 90] in the periodic setting converge in Gevrey norm exponentially fast to the uniform distribution as time $t\rightarrow \infty $.

Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc

In this paper we address the existence of time periodic solutions for the generalized inviscid SQG equation in the unit disc with homogeneous Dirichlet boundary condition when α∈(0,1). We show the existence of a countable family of bifurcating curves from the radial patches. In contrast with the preceding studies in active scalar equations, the Green function is no longer explicit and we circumvent this issue by a suitable splitting into a singular explicit part (which coincides with the planar one) and a smooth implicit one induced by the boundary of the domain. Another problem is connected to the analysis of the linear frequencies which admit a complicated form through a discrete sum involving Bessel functions and their zeros. We overcome this difficulty by using Sneddon's formula leading to a suitable integral representation of the frequencies.

Uniqueness for active scalar equations in a Zygmund space

We consider a class of active scalar equations which includes, for example, the 2D Euler equations, the 2D Navier–Stokes equations, and various aggregation equations including the Keller–Segel model. For this class of equations, we establish uniqueness of solutions in the Zygmund space C ∗ 0 . This result improves upon that in (Trans. Amer. Math. Soc. 367 (2015) 3095–3118), where the authors show uniqueness of solutions in BMO. As a corollary of our methods, we establish the uniform in space vanishing viscosity limit of Hölder continuous solutions to the aggregation equation with Newtonian potential.

Global solutions of aggregation equations and other flows with random diffusion

Aggregation equations, such as the parabolic-elliptic Patlak–Keller–Segel model, are known to have an optimal threshold for global existence versus finite-time blow-up. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. (Int Math Res Not IMRN 23:9370–9385, 2020) showing that, with high probability, the inviscid SQG equation with random diffusion has global classical solutions, we investigate whether suitable random diffusion can restore global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as those arising in aggregation models. For this class, we show global existence of solutions in Gevrey-type Fourier–Lebesgue spaces with quantifiable high probability.

On the blow up of a non-local transport equation in compact manifolds

In this note we show finite time blow-up for a class of non-local active scalar equations on compact Riemannian manifolds. The strategy we follow was introduced by Silvestre and Vicol [Trans. Amer. Math. Soc. 368 (2016), pp. 6159–6188] to deal with the one dimensional Córdoba-Córdoba-Fontelos equation and might be regarded as an instance of De Giorgi’s method.

On the analyticity of the trajectories of the particles in the planar patch problem for some active scalar equations

&lt;p style='text-indent:20px;'&gt;We prove analyticity in time of the particle trajectories associated with the solutions of some transport equations when the initial condition is the characteristic function of a regular bounded domain. These results are obtained from a detailed study of the Beurling transform, that represents a derivative of the velocity field. The precise estimates obtained for the solutions of an equation satisfied by the Lagrangian flow, are a key point in the development.&lt;/p&gt;

Symmetry in stationary and uniformly rotating solutions of active scalar equations

We study the radial symmetry properties of stationary and uniformly rotating solutions of the 2D Euler and gSQG equations, both in the smooth setting and the patch setting. For the 2D Euler equation, we show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial, without any assumptions on the connectedness of the support or the level sets. For the 2D Euler equation in the patch setting, we show that every uniformly rotating patch D with angular velocity Ω≤0 or Ω≥1 2 must be radial, where both bounds are sharp. For the gSQG equation, we obtain a similar symmetry result for Ω≤0 or Ω≥Ωα (with the bounds being sharp), under the additional assumption that the patch is simply connected. These results settle several open questions posed by Hmidi, de la Hoz, Hassainia, and Mateu on uniformly rotating patches. Along the way, we close a question by Choksi, Neumayer, and Topaloglu on overdetermined problems for the fractional Laplacian, which may be of independent interest. The main new ideas come from a calculus-of-variations point of view.

Vortex Patches Choreography for Active Scalar Equations

This paper deals with the existence of $N$ vortex patches located at the vertex of a regular polygon with $N$ sides that rotate around the center of the polygon at a constant angular velocity. That is done for Euler and (SQG)$_\beta$ equations, with $\beta\in(0,1)$, but may be also extended to more general models. The idea is the desingularization of the Thomsom polygon for the $N$ point vortex system, that is, $N$ point vortices located at the vertex of a regular polygon with $N$ sides. The proof is based on the study of the contour dynamics equation combined with the application of the infinite dimensional Implicit Function theorem and the well--chosen of the function spaces.

Finite-time singularity formation for an active scalar equation

We introduce an active scalar equation with a similar structure to the 3D Euler equations. Through studying the behavior of scale-invariant solutions, we show that compactly supported Lipschitz solutions belonging to can become singular in finite time. The interesting feature here is that we can achieve this in the absence of spatial boundaries.

Vanishing Diffusion Limits and Long Time Behaviour of a Class of Forced Active Scalar Equations

We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and one as the prefactor of the fractional Laplacian. We obtain existence and convergence results in certain parameter regimes and limits. We study the long time behaviour of solutions to the general problem and prove the existence of a unique global attractor. We apply results to two particular active scalar equations arising in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.

On the Non-diffusive Magneto-Geostrophic Equation

Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In Friedlander and Vicol (Nonlinearity 24(11)::3019–3042, 2011), the authors prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally well posed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stable for periodic perturbations with initial data localized in frequency straight lines crossing the origin. For such well-prepared data, the local existence and uniqueness of solutions can be obtained in Sobolev spaces and the global existence holds under a size condition over the $$H^{5/2^{+}}({{\mathbb {T}}}^3)$$ norm of the perturbation.

Strong ill-posedness of logarithmically regularized 2D Euler equations in the borderline Sobolev space

Logarithmically regularized 2D Euler equations are active scalar equations with the non-local velocity u=∇⊥Δ−1Tγω for the scalar ω. Two types of the regularizing operator Tγ with a parameter γ>0 are considered: Tγ=ln−γ⁡(e+|∇|) and Tγ=ln−γ⁡(e−Δ). These models regularize the 2D Euler equation for the vorticity (conventionally corresponding to the γ=0 case), which results in their local well-posedness in the borderline Sobolev space H1(R2)∩H˙−1(R2) when γ>12. In this paper, we examine the regularized models in the remaining regime γ≤12 and establish the strong ill-posedness in the borderline space. This completely solves the well-posedness problem of the regularized models in the borderline space by closing the gap between the local well-posedness result for γ>12 and the strong ill-posedness for γ=0.

Fractional Stochastic Active Scalar Equations Generalizing the Multi-Dimensional Quasi-Geostrophic &amp; 2D-Navier–Stokes Equations: The General Case

We introduce and study the well posedness: global existence, uniqueness and regularity of the solutions, of a class of d-dimensional fractional stochastic active scalar equations defined either on the torus $$ {\mathbb {T}}^d$$ or $$ {\mathbb {R}}^d$$ or $$ O\subsetneq {\mathbb {R}}^d$$ bounded, $$ d\ge 1$$ . This class includes, among others the stochastic; dD-quasi-geostrophic equation, fractional Burgers equation, fractional nonlocal transport equation and the 2D-fractional vorticity Navier–Stokes equation. We consider a locally Lipschitz diffusion term and a cylindrical Q-Wiener process with finite trace covariance Q. In particular, for $$ O={\mathbb {T}}^d$$ or $$ O\subsetneq {\mathbb {R}}^d$$ bounded, we prove the existence and the uniqueness of a global mild solution for the free divergence mode in the subcritical regime ( $$\alpha >\alpha _0(d)\ge 1$$ ), martingale solutions in the general regime ( $$\alpha \in (0, 2)$$ , supercritical, critical and subcritical) and free divergence mode and a local mild solution for the general mode and subcritical regime. Different kinds of regularity are also established for these solutions. For $$ O={\mathbb {R}}^d$$ , we studied the subcritical regime and we proved the existence of a global martingale solution for the free divergence mode and a local mild solution for the general mode.

Smooth solutions of the surface semi-geostrophic equations

The semi-geostrophic equations have attracted the attention of the physical and mathematical communities since the work of Hoskins in the 1970s owing to their ability to model the formation of fronts in rotation-dominated flows, and also to their connection with optimal transport theory. In this paper, we study an active scalar equation, whose activity is determined by way of a Neumann-to-Dirichlet map associated to a fully nonlinear second-order Neumann boundary value problem on the infinite strip $${\mathbb {R}}^2\times (0,1)$$, that models a semi-geostrophic flow in regime of constant potential vorticity. This system is an expression of an Eulerian semi-geostrophic flow in a coordinate system originally due to Hoskins, to which we shall refer as Hoskins’ coordinates. We obtain results on the local-in-time existence and uniqueness of classical solutions of this active scalar equation in Holder spaces.

On geometric and analytic mixing scales: comparability and convergence rates for transport problems

In this article we are interested in the geometric and analytic mixing scales of solutions to passive scalar problems. Here, we show that both notions are comparable after possibly removing large scale projections. In order to discuss our techniques in a transparent way, we further introduce a dyadic model problem. In a second part of our article we consider the question of sharp decay rates for both scales for Sobolev regular initial data when evolving under the transport equation and related active and passive scalar equations. Here, we show that slightly faster rates than the expected algebraic decay rates are optimal.

Wellposedness and Convergence of Solutions to a Class of Forced Non-diffusive Equations with Applications

This paper considers a family of non-diffusive active scalar equations where a viscosity type parameter enters the equations via the constitutive law that relates the drift velocity with the scalar field. The resulting operator is smooth when the viscosity is present but singular when the viscosity is zero. We obtain Gevrey-class local well-posedness results and convergence of solutions as the viscosity vanishes. We apply our results to two examples that are derived from physical systems: firstly a model for magnetostrophic turbulence in the Earth's fluid core and secondly flow in a porous media with an "effective viscosity".

Renormalization of active scalar equations

We consider transport equations with an incompressible transporting vector field. Whereas smooth solutions of such equations conserve every Lp norm simply by the chain rule, the question arises how regular a weak solution needs to be to guarantee this conservation property. Whereas the classical DiPerna–Lions theory gives sufficient conditions in terms of the regularity of the coefficients, with no regularity requirement on the transported scalar, we give here sufficient conditions in terms of the combined regularities of the coefficients and the scalar. This is motivated by the case of active scalar equations, where the transporting vector field has the same regularity as the transported scalar. We use commutator estimates similar to those of Constantin–E–Titi in the context of Onsager’s conjecture, but we require novel arguments to handle the case of Lp norms when p≠2.