Well-posedness of logarithmic spiral vortex sheets

We consider a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vortr\"age aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We prove that for each such spiral the normal component of the velocity field remains continuous across the spiral. Moreover, we give a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we show that a spiral gives rise to such a solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922. Another consequence of the main result is well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane.Comment: 25 pages, 3 figure

Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2

In this paper we prove finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. This is done in case of nonlinear diffusion and also in the case of nonlinear cross-diffusion provided the nonlinear chemosensitivity term is assumed not to decay. Moreover, it is shown that the above-mentioned lack of non-decay assumption is essential with respect to keeping the dichotomy finite-time blowup against boundedness of solutions. Namely, we prove that without the non-decay assumption possible asymptotic behaviour of solutions includes also infinite-time blowup.Comment: 14 page

Well-posedness for a model of individual clustering

25 pagesInternational audienceWe study the well-posedness of a model of individual clustering. Given p > N ≥ 1 and an initial condition in W 1,p (Ω), the local existence and uniqueness of a strong solution is proved. We next consider two specific reproduction rates and show global existence if N = 1, as well as, the convergence to steady states for one of these rates

Adenosine A2A receptors in Parkinson’s disease treatment

Latest results on the action of adenosine A2A receptor antagonists indicate their potential therapeutic usefulness in the treatment of Parkinson’s disease. Basal ganglia possess high levels of adenosine A2A receptors, mainly on the external surfaces of neurons located at the indirect tracts between the striatum, globus pallidus, and substantia nigra. Experiments with animal models of Parkinson’s disease indicate that adenosine A2A receptors are strongly involved in the regulation of the central nervous system. Co-localization of adenosine A2A and dopaminergic D2 receptors in striatum creates a milieu for antagonistic interaction between adenosine and dopamine. The experimental data prove that the best improvement of mobility in patients with Parkinson’s disease could be achieved with simultaneous activation of dopaminergic D2 receptors and inhibition of adenosine A2A receptors. In animal models of Parkinson’s disease, the use of selective antagonists of adenosine A2A receptors, such as istradefylline, led to the reversibility of movement dysfunction. These compounds might improve mobility during both monotherapy and co-administration with L-DOPA and dopamine receptor agonists. The use of adenosine A2A receptor antagonists in combination therapy enables the reduction of the L-DOPA doses, as well as a reduction of side effects. In combination therapy, the adenosine A2A receptor antagonists might be used in both moderate and advanced stages of Parkinson’s disease. The long-lasting administration of adenosine A2A receptor antagonists does not decrease the patient response and does not cause side effects typical of L-DOPA therapy. It was demonstrated in various animal models that inhibition of adenosine A2A receptors not only decreases the movement disturbance, but also reveals a neuroprotective activity, which might impede or stop the progression of the disease. Recently, clinical trials were completed on the use of istradefylline (KW-6002), an inhibitor of adenosine A2A receptors, as an anti-Parkinson drug