Agradecimentos: Abreu was partially supported by CNPq 306385/2019-8, BR and Ferreira was partially supported by FAPESP and CNPq, BR. This research is part of the doctoral thesis of Juan Gabriel Galeano Delgado, funded by CAPES (Coordination for the Improvement of Higher Education Personnel, Finance Code 001),which J. Galeano would like to point out and also thank the institutional support provided by IMECC/Unicamp Ver menos

Abstract: We study global well-posedness and finite time blow-up of solutions for a nonlinear one-dimensional transport equation with nonlocal velocity u(t) -(H(u)u)(x) = nu u(xx), nu > 0, and measure initial data. Such model arises in fluid mechanics in vortex-sheet problems and its nonlocal feature comes from the presence of a singular integral operator (Hilbert transform H(u)) in the velocity field. In the viscous case nu > 0, we analytically obtain an explicit condition on the size of the initial data for the global well-posedness in the framework of pseudomeasure spaces. In fact, we can give the condition depending on the initial-mass and analyze how the flow evolves from singular measures. Also, we numerically study blow-up of concentration type and global diffusion-smooth behavior of solutions. We obtain numerics that indicate the threshold value 2 pi for the initial-data mass that decides between blow-up or global smoothness of solutions. Such value is the same obtained for regular initial-data and by means of entropy methods. Thus, it seems to be intrinsic to the nonlocal PDE and independent of a particular framework, approach and initial-data regularity. The inviscid case nu = 0 is remarkable: simulations for model u(t) - (H(u)u)(x) = 0, evidence that the solution presents blow-up of concentration type with mass-preserving, while an attenuation effect is observed for the model with opposite sign in the nonlinearity u(t) + (H(u)u)(x) = 0, for any nontrivial (positive) measure as initial data Ver menos