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Gantumur Tsogtgerel

Publications and source records attributed to Gantumur Tsogtgerel.

13 recordsLinked to original sources
arXiv2026

Optimal Control in Hilbert Complex Spaces with Finite Element Exterior Calculus

We develop a framework for PDE-constrained optimal control on Hilbert complexes and its structure-preserving discretization by finite element exterior calculus (FEEC), with emphasis on problems where nontrivial topology creates physically meaningful global modes. We separate the gauge-fixed local state from harmonic components representing global circulation or flux, and introduce finite-dimensional topological actuators whose reachable modes are determined jointly by domain topology and actuator rank. We establish well-posedness, weak optimality conditions, reachability results, and FEEC error estimates preserving the harmonic structure. Numerical experiments on contractible and multiply connected domains demonstrate circulation and cavity- flux control, rank-dependent reachability, mesh-independent optimization, and topology-induced singular convergence. We further extend the framework to time-dependent Maxwell control, where harmonic electric circulation and magnetic flux become dynamical states exhibiting explicit conservation and reachability properties.

math.OC↗
arXiv2025

Regularity of solutions of the Navier-Stokes-αβ equations with wall-eddy boundary conditions

We establish global well-posedness and regularity for the Navier-Stokes-αβ system endowed with the wall-eddy boundary conditions proposed by Fried and Gurtin (2008). These conditions introduce a tangential vorticity traction proportional to wall vorticity and provide a continuum-mechanical model for near-wall turbulence. Our analysis begins with a variational formulation of the stationary fourth-order system, where we prove symmetry and a Gårding inequality for the associated bilinear form. We then verify Douglis-Nirenberg ellipticity and the Lopatinskii-Shapiro covering condition, establishing full Agmon-Douglis-Nirenberg regularity for the coupled system. Building on this framework, we derive a hierarchy of energy estimates for the nonlinear evolution equation, which yields global regularity, uniqueness, and stability. To our knowledge, this provides the first complete analytical treatment of the wall-eddy boundary model of Fried and Gurtin.

math.AP↗
arXiv2023

A Scaling Approach to Elliptic Theory for Geometrically-Natural Differential Operators with Sobolev-Type Coefficients

We develop local elliptic regularity for operators having coefficients in a range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij, Triebel-Lizorkin, Besov) where the coefficients have a regularity structure typical of operators in geometric analysis. The proofs rely on a nonstandard technique using rescaling estimates and apply to operators having coefficients with low regularity. For each class of function space for an operator's coefficients, we exhibit a natural associated range of function spaces of the same type for the domain of the operator and we provide regularity inference along with interior estimates. Additionally, we present a unified set of multiplication results for the function spaces we consider.

math.AP↗
arXiv2022

On uniqueness of weak solutions to the second boundary value problem for generated prescribed Jacobian equations

We prove that two Aleksandrov solutions of a generated prescribed Jacobian equation have the same gradients at points where they are both differentiable. For the optimal transportation case where two solutions can be translated to agree at a point without changing the $g$-subdifferential at that point, we recover the uniqueness up to a constant of solutions. For the general case, our result is a new proof with less regularity assumptions of a key theorem recently used to prove the uniqueness of solutions.

math.AP↗
arXiv2022

A prescribed scalar and boundary mean curvature problem and the Yamabe classification on asymptotically Euclidean manifolds with inner boundary

We consider the problem of finding a metric in a given conformal class with prescribed non-positive scalar curvature and non-positive boundary mean curvature on an asymptotically Euclidean manifold with inner boundary. We obtain a necessary and sufficient condition in terms of a conformal invariant of the zero sets of the target curvatures for the existence of solutions to the problem and use this result to establish the Yamabe classification of metrics in those manifolds with respect to the solvability of the prescribed curvature problem.

math.AP↗
arXiv2018

Convergence of Discrete Exterior Calculus Approximations for Poisson Problems

Discrete exterior calculus (DEC) is a framework for constructing discrete versions of exterior differential calculus objects, and is widely used in computer graphics, computational topology, and discretizations of the Hodge-Laplace operator and other related partial differential equations. However, a rigorous convergence analysis of DEC has always been lacking; as far as we are aware, the only convergence proof of DEC so far appeared is for the scalar Poisson problem in two dimensions, and it is based on reinterpreting the discretization as a finite element method. Moreover, even in two dimensions, there have been some puzzling numerical experiments reported in the literature, apparently suggesting that there is convergence without consistency. In this paper, we develop a general independent framework for analyzing issues such as convergence of DEC without relying on theories of other discretization methods, and demonstrate its usefulness by establishing convergence results for DEC beyond the Poisson problem in two dimensions. Namely, we prove that DEC solutions to the scalar Poisson problem in arbitrary dimensions converge pointwise to the exact solution at least linearly with respect to the mesh size. We illustrate the findings by various numerical experiments, which show that the convergence is in fact of second order when the solution is sufficiently regular. The problems of explaining the second order convergence, and of proving convergence for general p-forms remain open.

math.NA↗
arXiv2014

Axisymmetric critical points of a nonlocal isoperimetric problem on the two-sphere

On the two dimensional sphere, we consider axisymmetric critical points of an isoperimetric problem perturbed by a long-range interaction term. When the parameter controlling the nonlocal term is sufficiently large, we prove the existence of a local minimizer with arbitrary many interfaces in the axisymmetric class of admissible functions. These local minimizers in this restricted class are shown to be critical points in the broader sense (i.e., with respect to all perturbations). We then explore the rigidity, due to curvature effects, in the criticality condition via several quantitative results regarding the axisymmetric critical points.

math.CA↗
arXiv2014

On the consistency of the combinatorial codifferential

In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense that for a smooth enough differential form the combinatorial codifferential of the associated cochain converges to the exterior codifferential of the form as the triangulation is refined. In 1991, Smits proved this to be the case for the combinatorial codifferential applied to 1-forms in two dimensions under the additional assumption that the initial triangulation is refined in a completely regular fashion, by dividing each triangle into four similar triangles. In this paper we extend Smits's result to arbitrary dimensions, showing that the combinatorial codifferential on 1-forms is consistent if the triangulations are uniform or piecewise uniform in a certain precise sense. We also show that this restriction on the triangulations is needed, giving a counterexample in which a different regular refinement procedure, namely Whitney's standard subdivision, is used. Further, we show by numerical example that for 2-forms in three dimensions, the combinatorial codifferential is not consistent even for the most regular subdivision process.

math.DG↗
arXiv2013

The Lichnerowicz equation on compact manifolds with boundary

In this article we initiate a systematic study of the well-posedness theory of the Einstein constraint equations on compact manifolds with boundary. This is an important problem in general relativity, and it is particularly important in numerical relativity, as it arises in models of Cauchy surfaces containing asymptotically flat ends and/or trapped surfaces. Moreover, a number of technical obstacles that appear when developing the solution theory for open, asymptotically Euclidean manifolds have analogues on compact manifolds with boundary. As a first step, here we restrict ourselves to the Lichnerowicz equation, also called the Hamiltonian constraint equation, which is the main source of nonlinearity in the constraint system. The focus is on low regularity data and on the interaction between different types of boundary conditions, which has not been carefully analyzed before. In order to develop a well-posedness theory that mirrors the existing theory for the case of closed manifolds, we first generalize the Yamabe classification to nonsmooth metrics on compact manifolds with boundary. We then extend a result on conformal invariance to manifolds with boundary, and prove a uniqueness theorem. Finally, by using the method of sub- and super-solutions (order-preserving map iteration), we establish several existence results for a large class of problems covering a broad parameter regime, which includes most of the cases relevant in practice.

gr-qc↗
arXiv2010

Local Convergence of Adaptive Methods for Nonlinear Partial Differential Equations

In this article we develop convergence theory for a general class of adaptive approximation algorithms for abstract nonlinear operator equations on Banach spaces, and use the theory to obtain convergence results for practical adaptive finite element methods (AFEM) applied to several classes of nonlinear elliptic equations. In the first part of the paper, we develop a weak-* convergence framework for nonlinear operators, whose Gateaux derivatives are locally Lipschitz and satisfy a local inf-sup condition. The framework can be viewed as extending the recent convergence results for linear problems of Morin, Siebert and Veeser to a general nonlinear setting. We formulate an abstract adaptive approximation algorithm for nonlinear operator equations in Banach spaces with local structure. The weak-* convergence framework is then applied to this class of abstract locally adaptive algorithms, giving a general convergence result. The convergence result is then applied to a standard AFEM algorithm in the case of several semilinear and quasi-linear scalar elliptic equations and elliptic systems, including: a semilinear problem with subcritical nonlinearity, the steady Navier-Stokes equations, and a quasilinear problem with nonlinear diffusion. This yields several new AFEM convergence results for these nonlinear problems. In the second part of the paper we develop a second abstract convergence framework based on strong contraction, extending the recent contraction results for linear problems of Cascon, Kreuzer, Nochetto, and Siebert and of Mekchay and Nochetto to abstract nonlinear problems. The contraction result is then applied to a standard AFEM algorithm for semilinear problems with sub- and super-critical nonlinearities and for the Hamiltonian constraint in general relativity.

math.NA↗
arXiv2009

Analysis of a General Family of Regularized Navier-Stokes and MHD Models

We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. We give a unified analysis of the entire three-parameter family using only abstract mapping properties of the principle dissipation and smoothing operators, and then use specific parameterizations to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish results for singular perturbations, including the inviscid and alpha limits. Next we show existence of a global attractor for the general model, and give estimates for its dimension. We finish by establishing some results on determining operators for subfamilies of dissipative and non-dissipative models. In addition to establishing a number of results for all models in this general family, the framework recovers most of the previous results on existence, regularity, uniqueness, stability, attractor existence and dimension, and determining operators for well-known members of this family.

math.AP↗
arXiv2008

Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions

We consider the conformal decomposition of Einstein's constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of non-CMC weak solutions using a combination of a priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions for the Hamiltonian constraint, and topological fixed-point arguments. An important new feature of these results is the absense of the near-CMC assumption when the rescaled background metric is in the positive Yamabe class, if the freely specifiable part of the data given by the matter fields (if present) and the traceless-transverse part of the rescaled extrinsic curvature are taken to be sufficiently small. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, giving what are apparently the first non-CMC existence results without the near-CMC assumption. Standard bootstrapping arguments to increase the regularity of the conformal factor are blocked by the use of a weak background metric. In the CMC case, we recover Maxwell's rough solution results as a special case. Our results extend the 1996 non-CMC result of Isenberg and Moncrief in three ways: (1) the near-CMC assumption is removed in the case of the positive Yamabe class; (2) regularity is extended down to the maximum allowed by the background metric and the matter; and (3) the result holds for all three Yamabe classes. This last extension was also accomplished recently by Allen, Clausen and Isenberg, although their result is restricted to the near-CMC case and to smoother background metrics and data.

gr-qc↗