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arXiv · 2601.22611

Local controllability of the Cahn-Hilliard-Burgers' equation around certain steady states

Manika Bag·Sheetal Dharmatti·Subrata Majumdar·Debanjana Mitra
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Abstract

In this article we study the local controllability of the one-dimensional Cahn-Hilliard-Navier-Stokes equation, that is Cahn-Hilliard-Burgers' equation, around a certain steady state using a localized interior control acting only in the concentration equation. To do it, we first linearize the nonlinear equation around the steady state. The linearized system turns out to be a system coupled between second order and fourth order parabolic equations and the control acts in the fourth order parabolic equation. The null controllability of the linearized system is obtained by a duality argument proving an observability inequality. To prove the observability inequality, a new Carleman inequality for the coupled system is derived. Next, using the source term method, it is shown that the null controllability of the linearized system with non-homogeneous terms persists provided the non-homogeneous terms satisfy certain estimates in a suitable weighted space. Finally, using a Banach fixed point theorem in a suitable weighted space, the local controllability of the nonlinear system is obtained.

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Manika Bag, Sheetal Dharmatti, Subrata Majumdar, Debanjana Mitra. 2026-01-30. Local controllability of the Cahn-Hilliard-Burgers' equation around certain steady states. https://arxiv.org/abs/2601.22611

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