Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.15584

Homogenization of an optimal control problem for a Navier-Stokes-Cahn-Hilliard system in a 3D perforated domain

Amartya Chakrabortty·Haradhan Dutta·Hari Shankar Mahato
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the homogenization of an optimal control problem governed by a Navier-Stokes-Cahn-Hilliard system in a periodically perforated three-dimensional domain. The obstacles are Lipschitz sets of size $\varepsilon^α$, $α>3$, where $\varepsilon$ is the period. The fluid satisfies the no-slip condition on the outer boundary and the free-slip condition on the obstacles, the phase-field equation contains a source term, and the convective term carries a pore-scale Reynolds number that tends to zero as $\varepsilon\to0$. The control is a distributed body force, and the cost functional tracks the velocity in $L^2$ and the order parameter in $H^1$. Since uniqueness of weak solutions of the state system is not known for fixed $\varepsilon$, the control-to-state relation is multivalued, and we pose the microscopic problems over control-state pairs. We prove the existence of microscopic optimal pairs and the sequential $Γ$-convergence of the cost functionals to the cost functional of a control problem governed by the unsteady Stokes-Cahn-Hilliard system on the full domain. As a consequence, the optimal values converge, every cluster point of microscopic optimal pairs is optimal for the homogenized problem, and the optimal controls converge strongly along the corresponding subsequences. The $\liminf$ inequality follows from the homogenization of the state system, including the strong convergence of the gradients of the order parameter. Only the recovery sequence requires uniqueness of weak solutions of the homogenized system, which we prove when the outer boundary is of class $C^{2,1}$.

Explore related subjects

math.APmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Amartya Chakrabortty, Haradhan Dutta, Hari Shankar Mahato. 2026-10-11. Homogenization of an optimal control problem for a Navier-Stokes-Cahn-Hilliard system in a 3D perforated domain. https://arxiv.org/abs/2610.15584

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Suppression of blow-up in the 3D Patlak-Keller-Segel-Navier-Stokes system via non-parallel shear flows

In this paper, we consider the three-dimensional Patlak-Keller-Segel system coupled with the Navier-Stokes equations near the non-parallel shear flow $( Ay, 0, Ay )$ in $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. We show that if the shear flow is sufficiently strong (A is large enough), then the solutions to Patlak-Keller-Segel-Navier-Stokes system are global in time without any smallness restriction on the initial cell mass as long as the initial velocity satisfies $A^{\frac{2}{3}}\|u_{\rm in}\|_{H^2}\leq C_0$, which seems to be the first result of studying the suppression effect of shear flows for the 3D Patlak-Keller-Segel-Navier-Stokes system. Moreover, it implies that the solutions of the 3D Navier-Stokes equations are global in time if the initial velocity satisfies $A^{\frac{2}{3}}\|v_{\rm in}-(Ay,0,Ay)\|_{H^2}\leq C_0,$ which also shows the transition threshold for the shear flow $(Ay,0,Ay)$ in $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$.

math.AP↗
arXiv2026

Quantization analysis of Moser-Trudinger equations in the Poincaré disk and applications

In this paper, we first establish the quantitative properties for positive solutions to the Moser-Trudinger equation in the two-dimensional Poincaré disk $\mathbb{B}^2$: \begin{equation*}\label{mt1} \left\{ \begin{aligned} &-Δ_{\mathbb{B}^2}u=λue^{u^2},\ x\in\mathbb{B}^2, &u\to0,\ \text{when}\ ρ(x)\to\infty, &\|\nabla_{\mathbb{B}^2} u\|_{L^2(\mathbb{B}^2)}^2\leq M_0, \end{aligned} \right. \end{equation*} where $0<λ<\frac{1}{4}=\inf\limits_{u\in W^{1,2}(\mathbb{B}^2)\backslash\{0\}}\frac{\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2}{\|u\|_{L^2(\mathbb{B}^2)}^2}$, $ρ(x)$ denotes the geodesic distance between $x$ and the origin and $M_0$ is a fixed large positive constant (see Theorem \ref{thm1}). Furthermore, by carefully analyzing the asymptotic expansion of the Dirichlet energy \(\|\nabla_{\mathbb{B}^2} u\|_{L^2(\mathbb{B}^2)}^2\) as \(λ\to 0\), and establishing that \[ \lim_{\substack{λ\to 1/4}} \int_{\mathbb{B}^2} |\nabla_{\mathbb{B}^2} u_λ|^2 \, dV_{\mathbb{B}^2} = +\infty \] for~the ~nonzero~least energy solution, we are able to prove the existence of a critical point for the Moser-Trudinger functional \(F(u) = \int_{\mathbb{B}^2} (e^{u^2} - 1) \, dV_{\mathbb{B}^2}\) under the constraint \(\int_{\mathbb{B}^2} |\nabla_{\mathbb{B}^2} u|^2 \, dV_{\mathbb{B}^2} = Λ\) for \(Λ\in (4π, +\infty)\), up to some Möbius transformation. This result is substantially different from the non-existence of critical points for the Moser-Trudinger functional in the Euclidean disk. Finally, by applying a local Pohozaev identity around the origin and exploiting the uniqueness of solutions to the Cauchy initial value problem for ODEs, we prove that the Moser-Trudinger equation has a unique positive solution up to some Möbius transformation when $λ$ is close to $0$.

math.AP↗
arXiv2026

Local asymptotics for singular solutions to critical Hartree equations

For $n\geqslant3$ and $0<α<n$, we study positive classical solutions of critical Hartree equations that blow up at the origin. For the global problem on $\mathbb R^n\setminus\{0\}$, with the Hartree potential integrated over $\mathbb R^n$, we prove that solutions in $L^p(\mathbb R^n)$, where $p=(n+α)/(n-2)$, are radial and nonincreasing in $|x|$. We establish an integral representation and apply moving spheres to this nonlocal problem. For the local problem on $B_R\setminus\{0\}$, with the Hartree potential integrated over $B_R$, we prove $u(x)\leqslant C|x|^{-(n-2)/2}$ near the origin and \begin{equation*} u(x)=\bar u(|x|)\bigl(1+\mathcal O(|x|)\bigr)\qquad\text{as }x\to0, \end{equation*} where $\bar u$ is the spherical average of $u$. A main difficulty is controlling the nonlocal Hartree potential under Kelvin inversion and in the blow-up limit. The local proof combines blow-up analysis and moving spheres with estimates of the integral tails and an auxiliary Liouville lemma. We also give an alternative proof of the classification of positive $C^2$ solutions of the critical Hartree equation in $\mathbb R^n$.

math.AP↗