PDE Seminar
Speaker: Pardeep Sharma
Host: Harsha Hutridurga
Title: Impulsive inverse problem for the Navier-Stokes equations.
Time, day and date: 10:00:00 AM – 11:00:00 AM, Friday, May 22
Venue: Online (https://zoom.us/j/95223583387?pwd=i49UhXlwzHiLr6aLK6tqnlesWKIc8C.1)
Abstract: Attached (https://drive.google.com/file/d/1R6eEC2IhOQldZlImzdyH2z3OfWwTYANF/view?usp=sharing)

PDE Seminar
Speaker: Dr. Rakib Mondal
Host: Sivaji Ganesh Sista
Title: Rarefaction Wave Interaction and Existence of Global Solutions for a Hyperbolic System of PDEs
Time, day and date: 11:30:00 AM – 12:30:00 PM, Friday, May 22
Venue: Online (https://meet.google.com/ika-sdnw-yuf)
Abstract: In this talk, we discuss the first-order hyperbolic systems of partial differential equations and their mathematical challenges due to lack of regularity. In particular, we introduce a simplified 1-D system of balance laws governing blood flow with time-dependent body force. We discuss the interaction of two centered rarefaction waves and show that the interaction gives rise to a Goursat boundary value problem (GBVP). By transforming the system into Riemann-invariant coordinates, we discuss the existence and uniqueness of a global $C^1$ solution to the GBVP.

We further extend this system of balance laws considering both time-space dependent body force and the viscosity resistance, and prove the existence and stability of global weak entropy solutions. We establish a priori uniform $L^\infty$ bounds for a sequence of solutions to the regularized two-parameter viscosity-flux approximated systems. Using the vanishing-diffusion limit and the compensated compactness framework, we show the strong convergence and establish the global weak entropy solutions to the original Cauchy problem with large initial data.

IPDF Talk
Speaker: Rahul Bhardwaj, IIT Ropar
Host: Suman Kumar Sahoo
Title: Reconstruction Results for Inverse Problems in Integral Geometry and Hyperbolic PDEs
Time, day and date: 4:00:00 PM – 5:00:00 PM, Friday, May 22
Venue: Online (https://zoom.us/launch/jc/93166883629) (Meeting ID: 931 6688 3629 Passcode: 506739)
Abstract: In this talk, we address two inverse problems arising in integral geometry and hyperbolic partial differential equations (PDEs). The first problem concerns the reconstruction of a vector field in the plane from a set of generalised V-line transforms, namely longitudinal and transverse V-line transforms. Specifically, we focus on recovering a vector field which is supported inside a disk of radius R centered at the origin. This recovery process is accomplished through a combination of the aforementioned generalized V-line transforms, using two different techniques for different sets of data. The second problem concerns an inverse problem for a semi-linear wave equation posed on a bounded subset of Euclidean space. The objective is to reconstruct the coefficients from the associated Dirichlet-to-Neumann map. The analysis is based on the method of higher-order linearization.