arXiv subjects

Mansur Shakipov

Publications and source records attributed to Mansur Shakipov.

6 recordsLinked to original sources

arXiv2026

Existence of weak solutions of the surface Beris-Edwards model

We prove the existence of weak solutions to the surface Beris-Edwards model for nematic liquid crystals posed on a $d$-dimensional ($d \in \{2,3\}$) closed hypersurface of class $C^{2,1}$. This thermodynamically consistent model, recently introduced by Bouck, Nochetto and Yushutin (2024), couples the incompressible tangent Navier-Stokes equations with a kinematic equation for the Q-tensor field that encodes the orientation of the liquid crystal particles with a general state of orientational order. Extending ideas by Abels, Dolzmann and Liu (2014) and Guillén-González and Rodríguez-Bellido (2015) for the Beris-Edwards model in flat domains, we design a Faedo-Galerkin scheme based upon eigenfunctions of an appropriate tangent Stokes operator and tensor-valued Laplace-Beltrami operator and recover a weak solution via standard compactness arguments.

math.AP↗

arXiv2026

Ferrofluids: Modeling and Approximation

Starting from Maxwell's and linear momentum balance equations, we derive a ferrofluid model using the generalized Onsager's principle. Guided by a discrete perturbation estimate, we design and analyze families of Galerkin schemes that converge to sufficiently regular solutions and derive error estimates. Finally, we numerically explore the model with our proposed method.

math.NA↗

arXiv2026

$\mathrm{L}^p$-based Sobolev theory on closed manifolds of minimal regularity: Vector-valued problems

This paper is the second part of a two-paper series, initiated in arXiv:2603.02163 for scalar PDEs on hypersurfaces, and is concerned with the well-posedness and $\mathrm{L}^p$-based Sobolev regularity of vector-valued PDEs of interest in fluid dynamics. This family of PDEs includes the (stationary) Bochner Laplace, tangent Stokes and Oseen, and tangent Navier--Stokes equations. We present several strong, weak and ultra-weak formulations of these problems on compact, connected $d$-dimensional manifolds without boundary embedded in $\mathrm{R}^{d+1}$. We prove $\mathrm{W}^{m,p}$-regularity for any $p \in (1,\infty)$ for manifolds of minimal regularity $C^{m+1}$ or $C^{m,1}$ for $m\ge1$. Building upon the $\mathrm{L}^p$-based scalar elliptic theory from arXiv:2603.02163, we develop a parametrization-free and purely variational approach that resorts to classical results such as the Banach--Nečas--Babuška theorem and the generalized Babuška--Brezzi theory in reflexive Banach spaces. In particular, by exploiting the manifold closedness, we decouple the velocity and pressure variables in the tangent Stokes problem to establish their higher-regularity $\mathbf{W}^{m,p} \times \mathrm{W}^{m-1,p}$ ($m \geq 2$) as a consequence of the $\mathrm{L}^p$-based well-posedness and regularity theory for the Laplace--Beltrami and Bochner--Laplace operators. We study spectral and regularity properties of an appropriate Stokes operator, and apply them to show existence of solutions for the Navier--Stokes equations for $p=2$ and $d \leq 4$. We next extend the well-posedness to $p > 2$ and prove higher-order $\mathrm{L}^p$-based regularity. We finally examine alternative choices to the Bochner Laplace operator that are useful in fluid dynamics.

math.AP↗

arXiv2026

$\mathrm{L}^p$-based Sobolev theory on closed manifolds of minimal regularity: Scalar Elliptic Equations

This paper and its follow-up arXiv:2508.11109 are concerned with the well-posedness and $\mathrm{L}^p$-based Sobolev regularity for appropriate weak formulations of a family of prototypical PDEs posed on manifolds of minimal regularity. In particular, the domains are assumed to be compact, connected $d$-dimensional manifolds without boundary of class $C^k$ and $C^{k-1,1}$ ($k \geq 1$) embedded in $\mathrm{R}^{d+1}$. The focus of this program is on the $\mathrm{L}^p$-based theory that is sharp with respect to the regularity of the source terms and the manifold. In the present paper, we focus our attention on the case of general scalar elliptic problems. We first establish $\mathrm{L}^p$-based well-posedness and higher regularity for the purely diffusive problems with variable coefficients by localizing and rewriting these equations in flat domains to employ the Calderón--Zygmund theory, combined with duality arguments. We then invoke the Fredholm alternative to derive analogous results for general scalar elliptic problems, underscoring the subtle differences that the geometric setting entails compared to the theory in flat domains.

math.AP↗

arXiv2025

Surface Stokes Without Inf-Sup Condition

For a $d$-dimensional hypersurface of class $C^3$ without boundary, we reformulate the surface Stokes equations as a nonsymmetric indefinite elliptic problem governed by two Laplacians. We then use this elliptic reformulation as a basis for a numerical method based on lifted parametric FEM. Assuming no geometric error for simplicity, we prove its well-posedness, quasi-best approximation in a robust mesh-dependent $H^1$-norm for any polynomial degree, as well as an optimal $L^2$ error estimate for both velocity and pressure. This entails a sufficiently small mesh size that solely depends on the Weingarten map and circumvents the usual discrete inf-sup condition. We present numerical experiments for velocity-pressure pairs with equal and disparate polynomial degrees, demonstrating that the proposed method is both accurate and practical.

math.NA↗

arXiv2024

Inf-Sup Stability of Parabolic TraceFEM

We develop a parabolic inf-sup theory for a modified TraceFEM semi-discretization in space of the heat equation posed on a stationary surface embedded in $\mathbb{R}^n$. We consider the normal derivative volume stabilization and add an $L^2$-type stabilization to the time derivative. We assume that the representation of and the integration over the surface are exact, however, all our results are independent of how the surface cuts the bulk mesh. For any mesh for which the method is well-defined, we establish necessary and sufficient conditions for inf-sup stability of the proposed TraceFEM in terms of $H^1$-stability of a stabilized $L^2$-projection and of an inverse inequality constant that accounts for the lack of conformity of TraceFEM. Furthermore, we prove that the latter two quantities are bounded uniformly for a sequence of shape-regular and quasi-uniform bulk meshes. We derive several consequences of uniform discrete inf-sup stability, namely uniform well-posedness, discrete maximal parabolic regularity, parabolic quasi-best approximation, convergence to minimal regularity solutions, and optimal order-regularity energy and $L^2 L^2$ error estimates. We show that the additional stabilization of the time derivative restores optimal conditioning of time-discrete TraceFEM typical of fitted discretizations.

math.NA↗