Title: Amplitude Blowup in Radial Euler Flows
Abstract: We show that the full compressible Euler system admits unbounded solutions. The examples are radial flows of similarity type and describe a spherically symmetric and continuous wave moving toward the origin. At time of focusing, the primary flow variables suffer amplitude blowup at the origin. The flow is continued beyond collapse and gives rise to an expanding shock wave. We verify that the resulting flow provides a genuine weak solution to the full, multi-dimensional compressible Euler system. While unbounded radial Euler flows have been known since the work of Guderley (1942), those are at the borderline of the regime covered by the Euler model: the upstream pressure field vanishes identically (either because of vanishing temperature or vanishing density there). In contrast, the solutions we build exhibit an everywhere strictly positive pressure field, demonstrating that the geometric effect of wave focusing is strong enough on its own to generate unbounded values of primary flow variables. This is joint work with Helge Kristian Jenssen (PSU).

Title: Microlocal control estimates for 0th order operators
Abstract: Internal waves are gravity waves in density stratified fluids. In 2D aquaria, Maas et al predicted and then observed in experiments the internal wave attractors. Colin de Verd\`iere–Saint-Raymond used 0th order pseudodifferential operators with Morse–Smale dynamical assumptions as microlocal models for internal waves. In this talk, I will show how the dynamical assumptions on the classical flow of 0th order operators allow us to obtain microlocal control estimates for 0th order operators.

Title: Application of Quasilinearization method to unsteady nanofluid flow problem
Abstract: The unsteady boundary layer flow of nanofluid in a free stream is investigated . A new self-similar solutions is obtained and the resulting system of nonlinear ordinary differential equations is solved numerically using an implicit finite difference scheme in combination with the quasilinearization method. Numerical results are presented for the skin friction coefficient, the local Nusselt number and the local Sherwood number as well as for the velocity, temperature and the nanoparticle volume fraction profiles.

Title: Superdiffusive Fractional In time NLS: Nonlinear Theory
Abstract: We consider a new class of partial differential equations dubbed Superdiffusive Fractional-In-Time Nonlinear Schrödinger equations on bounded domains. This equation will serve as a proper interpolation between the nonlinear Schrödinger equation and the Klein-Gordon Wave equation by making use of the (nonlocal in time) Caputo fractional time derivative of order $1<\beta< 2$. In space, we consider nonnegative $L^2(X)$ self-adjoint operators $A$ which include Laplacians and the (nonlocal in space) fractional Laplace operators with appropriate boundary conditions. In this talk, we present recent advances in the nonlinear theory, behavior of solutions, and potential applications as a control problem. Such models will serve appropriately in the important study of nonlinear optics through dispersive media and in the study of Bose-Einstein condensates.