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In this paper we characterize the minimizer for a class of nonlocal perturbations of the Coulomb energy. We show that the minimizer is the normalized characteristic function of an ellipsoid, under the assumption that the perturbation kernel has the same homogeneity as the Coulomb potential, is even, is smooth off the origin, and is sufficiently small. This result can be seen as the stability of ellipsoids as energy minimizers, since the minimizer of the Coulomb energy is the normalized characteristic function of a ball. [ABSTRACT FROM AUTHOR]

Background: One-catheter strategy, based in multipurpose catheters, allows exploring both coronary arteries with a single catheter. This strategy could simplify coronary catheterization and reduce the volume of contrast administration, by reducing radial spasm. To date, observational studies showed greater benefits regarding contrast consumption and catheterization performance than controlled trials. The aim of this work is to perform the first systematic review and meta-analysis of randomized clinical trials (RCT) to adequately quantify the benefits of one-catheter strategy, with multipurpose catheters, over conventional two-catheter strategy on contrast consumption, and catheterization performance. Methods: A search in PubMed, CINALH, and CENTRAL databases was conducted to identify randomized trials comparing one-catheter and two-catheter strategies. The primary outcome was volume of iodinated contrast administrated. Secondary endpoints, evaluating coronary catheterization performance included: arterial spasm, fluoroscopy time, and procedural time. Results: Five RCT were included for the final analysis, with a total of 1599 patients (802 patients with one-catheter strategy and 797 patients with two-catheter strategy). One-catheter strategy required less administration of radiological contrast (difference in means [DiM] [95% confidence interval (CI)]; -3.831 mL [-6.165 mL to-1.496 mL], p = 0.001) as compared to two-catheter strategy. Furthermore, less radial spasm (odds ratio [95% CI], 0.484 [0.363 to 0.644], p < 0.001) and less procedural time (DiM [95% CI], -72.471 s [-99.694 s to -45.249 s], p < 0.001) were observed in one-catheter strategy. No differences on fluoroscopy time were observed. Conclusions: One-catheter strategy induces a minimal reduction on radiological contrast administration but improves coronary catheterization performance by reducing arterial spasm and procedural time as compared to conventional two-catheter strategy. [ABSTRACT FROM AUTHOR]

In this paper we consider a general class of anisotropic energies in three dimensions and give a complete characterisation of their minimisers. We show that, depending on the Fourier transform of the interaction potential, the minimiser is either the normalised characteristic function of an ellipsoid or a measure supported on a two-dimensional ellipse. In particular, it is always an ellipsoid if the transform is strictly positive, while when the Fourier transform is degenerate both cases can occur. Finally, we show an explicit example where loss of dimensionality of the minimiser does occur. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the $${{\rm (SQG)}_{\alpha}}$$ equations with $${\alpha \in (0,1)}$$ . From the numerical experiments implemented for Euler equations in Deem and Zabusky (Phys Rev Lett 40(13):859-862, 1978), Pierrehumbert (J Fluid Mech 99:129-144, 1980), Saffman and Szeto (Phys Fluids 23(12):2339-2342, 1980) it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs. There are some analytical proofs based on variational principle (Keady in J Aust Math Soc Ser B 26:487-502, 1985; Turkington in Nonlinear Anal Theory Methods Appl 9(4):351-369, 1985); however, they do not give enough information about the pairs, such as the uniqueness or the topological structure of each single vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the $${{\rm (SQG)}_{\alpha}}$$ equation when $${\alpha \in (0,1)}$$ . The proofs rely on the contour dynamics equations combined with a desingularization of the point vortex pairs and the application of the implicit function theorem. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of doubly-connected rotating patches for Euler equations when the classical non-degeneracy conditions are not satisfied. We prove the bifurcation of the V-states with two-fold symmetry, however for higher m -fold symmetry with m ≥ 3 the bifurcation does not occur. This answers to a problem left open in [10] . Note that, contrary to the known results for simply-connected and doubly-connected cases where the bifurcation is pitchfork, we show that the degenerate bifurcation is actually transcritical. These results are in agreement with the numerical observations recently discussed in [10] . The proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation. [ABSTRACT FROM AUTHOR]

In this paper, we construct time periodic doubly connected solutions for the threedimensional quasi-geostrophic model in the patch setting. More specifically, we prove the existence of nontrivial m-fold doubly connected rotating patches bifurcating from a generic doubly connected revolution shape domain with higher symmetry m > mo and mo large enough. The linearized matrix operator at the equilibrium state has variable and singular coefficients and its spectral analysis is performed via the approach devised in [C. Garcia, T. Hmidi, and J. Mateu, Comm. Math. Phys., 390 (2022), pp. 617--756], where a suitable symmetrization was introduced. New difficulties emerge due to the interaction between the surfaces making the spectral problem richer and involved. [ABSTRACT FROM AUTHOR]

It is known that the improved Cotlar's inequality B∗f(z)≤CM(Bf)(z), z∈C, holds for the Beurling transform B, the maximal Beurling transform B∗f(z)= supε>0∣∫|w|>εf(z−w)1/w²dw∣, z∈C, and the Hardy-Littlewood maximal operator M. In this note we consider the maximal Beurling transform associated with squares, namely, B*Sf(z)=supε>0∣∫w∉Q(0,ε) f(z−w)1/w²dw∣∣∣, z∈C, Q(0,ε) being the square with sides parallel to the coordinate axis of side length ε. We prove that B*Sf(z)≤CM²(Bf)(z), z∈C, where M²=M ○ M is the iteration of the Hardy-Littlewood maximal operator, and M² cannot be replaced by M. [ABSTRACT FROM AUTHOR]

In this paper we consider rotating doubly connected vortex patches for the Euler equations in the plane. When the inner interface is an ellipse we show that the exterior interface must be a confocal ellipse. We then discuss some relations, first found by Flierl and Polvani, between the parameters of the ellipses, the velocity of rotation and the magnitude of the vorticity in the domain enclosed by the inner ellipse. [ABSTRACT FROM AUTHOR]

We show that the boundary of a rotating vortex patch (or V-state, in the terminology of Deem and Zabusky) is C, provided the patch is close to the bifurcation circle in the Lipschitz norm. The rotating patch is also convex if it is close to the bifurcation circle in the C norm. Our proof is based on Burbea's approach to V-states. [ABSTRACT FROM AUTHOR]

In this paper, we pursue the study of the problem of controlling the maximal singular integral T∗f by the singular integral Tf. Here T is a smooth homogeneous Calderón–Zygmund singular integral of convolution type. We consider two forms of control, namely, in the norm and via pointwise estimates of T∗f by M(Tf) or M2(Tf), where M is the Hardy–Littlewood maximal operator and its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article, we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type arises. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak (1,1) estimates, which are no longer true for odd kernels. For odd kernels, we obtain sharp weaker inequalities involving a weak L1 estimate for functions in L log L. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper we discuss a special class of Beltrami coefficients whose associated quasiconformal mapping is bilipschitz. A particular example are those of the form , where Ω is a bounded domain with boundary of class and f a function in satisfying . An important point is that there is no restriction whatsoever on the norm of f besides the requirement on Beltrami coefficients that the supremum norm be less than 1. The crucial fact in the proof is the extra cancellation enjoyed by even homogeneous Calderón–Zygmund kernels, namely that they have zero integral on half the unit ball. This property is expressed in a particularly suggestive way and is shown to have far reaching consequences. An application to a Lipschitz regularity result for solutions of second order elliptic equations in divergence form in the plane is presented. [Copyright &y& Elsevier]

This paper aims to study time periodic solutions for 3D inviscid quasi-geostrophic model. We show the existence of non trivial rotating patches by suitable perturbation of stationary solutions given by generic revolution shapes around the vertical axis. The construction of those special solutions are done through bifurcation theory. In general, the spectral problem is very delicate and strongly depends on the shape of the initial stationary solutions. More specifically, the spectral study can be related to an eigenvalue problem of a self-adjoint compact operator. We are able to implement the bifurcation only from the largest eigenvalues of the operator, which are simple. Additional difficulties generated by the singularities of the poles are solved through the use of suitable function spaces with Dirichlet boundary condition type and refined potential theory with anisotropic kernels. [ABSTRACT FROM AUTHOR]

Abstract We prove well-posedness of linear scalar conservation laws using only assumptions on the growth and the modulus of continuity of the velocity field, but not on its divergence. As an application, we obtain uniqueness of solutions in the atomic Hardy space, H 1 , for the scalar conservation law induced by a class of vector fields whose divergence is an unbounded BMO function. [ABSTRACT FROM AUTHOR]

In this note, we study the well-posedness of the Cauchy problem for the transport equation in the BMO space and certain Triebel–Lizorkin spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study flows associated to Sobolev vector fields with subexponentially integrable divergence. Our approach is based on the transport equation following DiPerna–Lions [17] . A key ingredient is to use a quantitative estimate of solutions to the Cauchy problem of transport equation to obtain the regularity of density functions. [ABSTRACT FROM AUTHOR]

We prove existence of doubly connected V -states for the planar Euler equations which are not annuli. The proof proceeds by bifurcation from annuli at simple eigenvalues. The bifurcated V -states we obtain enjoy an m-fold symmetry for some m ≥ 3. [ABSTRACT FROM AUTHOR]