Mathematics Colloquium

Date and time: 14 September 2022 at 4 pm

Venue: Ramanujan Hall (Room 213, II Floor)

Title:  Boundary null-controllability of 1d linearized compressible 
Navier-Stokes
System by one control force.

Speaker:  Shirshendu Chowdhury, Department of Mathematics and 
Statistics, IISER Kolkata,

Abstract: In the first part of the talk, we introduce the concept of 
controllability of Differential Equations. Then we give some examples in 
finite (ODE) and infinite dimensional(PDE) contexts. We recall the 
controllability results of the Transport and Heat equation.

In the second part of the talk, we consider compressible Navier-Stokes 
equations in one dimension, linearized around a constant steady state  
$(Q_0, V_0 ) $, with $Q_ 0 > 0, V 0 >0 $.  It is a Coupled system of 
transport and heat type equations.  We study the boundary 
null-controllability of this linearized system in the interval $(0,1)$ 
when a Dirichlet control function is acting either only on the density 
or only on the velocity component at one end of the interval.

We obtain null controllability using one boundary control in the space 
${H}^s_{per}(0,1)\times L^2(0,1)$ for any $s>\frac{1}{2}$ provided the 
time $T>1$, where ${H}_{per}^s(0,1)$ denotes the Sobolev space of 
periodic functions. The proof is based on spectral analysis and on 
solving a mixed parabolic-hyperbolic moments problem and a parabolic 
hyperbolic joint Ingham-type inequality.

This is a recent joint work (https://arxiv.org/abs/2204.02375 [1], 2022) 
with Kuntal Bhandari, Rajib Dutta and Jiten Kumbhakar.

Note: Mathematics Colloquia are accessible to the general audience

Links:
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[1] https://arxiv.org/abs/2204.02375