Partial Differential Equations Seminar
Speaker: Sanchit Chaturvedi, New York University
Host: Harsha Hutridurga
Title: Global problems in kinetic theory
Time, day and date: 4:00:00 PM – 5:00:00 PM, Wednesday, January 07
Venue: Ramanujan Hall
Abstract: In this talk I will discuss the problem of global existence and stability of special solutions to kinetic collisional equations that model the dynamics of gas particles. In the first half of the talk I will consider the problem of stability of global Maxwellians for the Landau equations (and Vlasov--Poisson--Landau for the charged case) on a torus and how collisions cause the global Maxwellians to be stable. In the second half I will focus on global problems on the whole space and discuss how the problem of stability is way richer in this case due to the complex interaction of entropic dissipation (due to collisions) and dispersion

Coagulation-fragmentation equations
Speaker: Ram Gopal Jaiswal, IIT Bombay
Host: Harsh Hutridurga
Title: A first course on the Coagulation-Fragmentation equations.
Time, day and date: 11:00:00 AM – 12:30:00 PM, Thursday, January 08
Venue: Ramanujan Hall
Abstract: This series of lectures focuses on the existence and uniqueness of weak solutions to the continuous coagulation equation under suitable assumptions on the coagulation kernel. We study separately the regimes in which solutions conserve total mass and those in which mass conservation may fail in finite time due to a phenomenon known as gelation. We then incorporate fragmentation processes, treating linear and nonlinear (collision-induced) fragmentation separately, and establish the corresponding existence and uniqueness results. Moreover, we prove the existence of mass-conserving stationary (equilibrium) solutions to coagulation equations with linear fragmentation under appropriate assumptions on the coagulation and fragmentation kernels

Talk
Speaker: Utsab Sarkar, IIT Bombay
Host: Mayukh Mukherjee
Title: Stability in Critical Variational Problems
Time, day and date: 5:00:00 PM – 6:00:00 PM, Thursday, January 08
Venue: Online (meet.google.com/gzx-igvq-thu)
Abstract: In the first part (joint work with Souptik Chakraborty), we study quantitative stability for the fractional Hardy–Sobolev inequality and prove a sharp Bianchi-Egnell-type stability estimate showing that near-equality implies proximity to the manifold of minimizers. We then analyze the associated Euler-Lagrange equation, establishing a qualitative Struwe decomposition for near-solutions via Palais-Smale analysis, and showing that finite bubbling is the only obstruction to compactness in all dimensions. Finally, in the low-dimensional range (2s<N<6s-2t), we develop a sharp quantitative multi-bubble theory in which the Euler-Lagrange deficit controls linearly the distance to the multi-bubble manifold and weak interactions between distinct bubbles.
In the second part (joint work with Mayukh Mukherjee), we discuss Dirichlet Gagliardo Nirenberg inequalities on compact manifolds with boundary. We identify the sharp Euclidean and half-space thresholds, establish quantitative stability via transfer from the model geometries, and describe the compactness–bubbling dichotomy at the threshold in terms of boundary geometry.