More Like this

1. Stability and Large-Time Behavior on 3D Incompressible MHD Equations with Partial Dissipation Near a Background Magnetic Field

   https://doi.org/10.1007/s00205-025-02100-4  

   Lin, Hongxia; Wu, Jiahong; Zhu, Yi (April 2025, Archive for Rational Mechanics and Analysis)

   Abstract Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and dampens electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in$$\mathbb R^3$$ $R^3$. The velocity equation in this system is the 3D Navier–Stokes equation with dissipation only in the$$x_1$$ $x_1$-direction, while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field (0, 1, 0) is globally stable in the Sobolev setting$$H^3({\mathbb {R}}^3)$$ $H^3\left(R^3\right)$. In addition, explicit decay rates in$$H^2({\mathbb {R}}^3)$$ $H^2\left(R^3\right)$are also obtained. For when there is no presence of a magnetic field, the 3D anisotropic Navier–Stokes equation is not well understood and the small data global well-posedness in$$\mathbb R^3$$ $R^3$remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps to stabilize the fluid. 

   more » « less
2. Two-Grid Stabilized Lowest Equal-Order Finite Element Method for the Dual-Permeability-Stokes Fluid Flow Model

   https://doi.org/10.1007/s10915-024-02723-x  

   Haque, Md_Nazmul; Nasu, Nasrin_Jahan; Al_Mahbub, Md_Abdullah; Mohebujjaman, Muhammad (November 2024, Journal of Scientific Computing)

   Abstract This paper proposes and investigates the two-grid stabilized lowest equal-order finite element method for the time-independent dual-permeability-Stokes model with the Beavers-Joseph-Saffman-Jones interface conditions. This method is mainly based on the idea of combining the two-grid and the two local Gauss integrals for the dual-permeability-Stokes system. In this technique, we use a difference between a consistent mass matrix and an under-integrated mass matrix for the pressure variable of the dual-permeability-Stokes model using the lowest equal-order finite element quadruples. In the two-grid scheme, the global problem is solved using the standard$$ P_1-P_1-P_1-P_1 $$ $P_1-P_1-P_1-P_1$finite element approximations only on a coarse grid with grid sizeH. Then, a coarse grid solution is applied on a fine grid of sizehto decouple the interface terms and the mass exchange terms for solving the three independent subproblems such as the Stokes equations, microfracture equations, and the matrix equations on the fine grid. On the other hand, microfracture and matrix equations are decoupled through the mass exchange terms. The weak formulation is reported, and the optimal error estimate is derived for the two-grid schemes. Furthermore, the numerical results validate that the two-grid stabilized lowest equal-order finite element method is effective and has the same accuracy as the coupling scheme when we choose$$ h=H^2 $$ $h=H^2$. 

   more » « less
3. The Transition from Darcy to Nonlinear Flow in Heterogeneous Porous Media: I—Single-Phase Flow

   https://doi.org/10.1007/s11242-024-02070-3  

   Arbabi, Sepehr; Sahimi, Muhammad (March 2024, Transport in Porous Media)

   Abstract Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by two-dimensional slices of a three-dimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient$${\varvec{\nabla }}P$$ $\nabla P$and the fluid velocityv. The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude$$\omega _z$$ ${\omega }_z$of the vorticity is nearly zero. As Re increases, however, so also does$$\omega _z$$ ${\omega }_z$, and its rise from nearly zero begins at the same Re at which the Darcy’s law breaks down. We also show that a nonlinear relation between the macroscopic pressure gradient and the fluid velocityv, given by,$$-{\varvec{\nabla }}P=(\mu /K_e)\textbf{v}+\beta _n\rho |\textbf{v}|^2\textbf{v}$$ $-\nabla P=(\mu /K_e)v+{\beta }_n\rho {\left|v\right|}^{2}v$, provides accurate representation of the numerical data, where$$\mu$$ $\mu$and$$\rho$$ $\rho$are the fluid’s viscosity and density,$$K_e$$ $K_e$is the effective Darcy permeability in the linear regime, and$$\beta _n$$ ${\beta }_n$is a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation. 

   more » « less
4. Comparing solution paths of sparse quadratic minimization with a Stieltjes matrix

   https://doi.org/10.1007/s10107-023-01966-0  

   He, Ziyu; Han, Shaoning; Gómez, Andrés; Cui, Ying; Pang, Jong-Shi (May 2023, Mathematical Programming)

   Abstract This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “$$\ell _0$$ ${\ell }_0$-path” where the discontinuous$$\ell _0$$ ${\ell }_0$-function provides the exact sparsity count; the “$$\ell _1$$ ${\ell }_1$-path” where the$$\ell _1$$ ${\ell }_1$-function provides a convex surrogate of sparsity count; and the “capped$$\ell _1$$ ${\ell }_1$-path” where the nonconvex nondifferentiable capped$$\ell _1$$ ${\ell }_1$-function aims to enhance the$$\ell _1$$ ${\ell }_1$-approximation. Serving different purposes, each of these three formulations is different from each other, both analytically and computationally. Our results deepen the understanding of (old and new) properties of the associated paths, highlight the pros, cons, and tradeoffs of these sparse optimization models, and provide numerical evidence to support the practical superiority of the capped$$\ell _1$$ ${\ell }_1$-path. Our study of the capped$$\ell _1$$ ${\ell }_1$-path is interesting in its own right as the path pertains to computable directionally stationary (= strongly locally minimizing in this context, as opposed to globally optimal) solutions of a parametric nonconvex nondifferentiable optimization problem. Motivated by classical parametric quadratic programming theory and reinforced by modern statistical learning studies, both casting an exponential perspective in fully describing such solution paths, we also aim to address the question of whether some of them can be fully traced in strongly polynomial time in the problem dimensions. A major conclusion of this paper is that a path of directional stationary solutions of the capped$$\ell _1$$ ${\ell }_1$-regularized problem offers interesting theoretical properties and practical compromise between the$$\ell _0$$ ${\ell }_0$-path and the$$\ell _1$$ ${\ell }_1$-path. Indeed, while the$$\ell _0$$ ${\ell }_0$-path is computationally prohibitive and greatly handicapped by the repeated solution of mixed-integer nonlinear programs, the quality of$$\ell _1$$ ${\ell }_1$-path, in terms of the two criteria—loss and sparsity—in the estimation objective, is inferior to the capped$$\ell _1$$ ${\ell }_1$-path; the latter can be obtained efficiently by a combination of a parametric pivoting-like scheme supplemented by an algorithm that takes advantage of the Z-matrix structure of the loss function. 

   more » « less
5. Rates of convergence for regression with the graph poly-Laplacian

   https://doi.org/10.1007/s43670-023-00075-5  

   Trillos, Nicolás García; Murray, Ryan; Thorpe, Matthew (November 2023, Sampling Theory, Signal Processing, and Data Analysis)

   Abstract In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset$$\{x_i\}_{i=1}^n$$ ${\left\{x_i\right\}}_{i=1}^n$and a set of noisy labels$$\{y_i\}_{i=1}^n\subset \mathbb {R}$$ ${\left\{y_i\right\}}_{i=1}^n\subset R$we let$$u_n{:}\{x_i\}_{i=1}^n\rightarrow \mathbb {R}$$ $u_n:{\left\{x_i\right\}}_{i=1}^n\to R$be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When$$y_i = g(x_i)+\xi _i$$ $y_i=g\left(x_i\right)+{\xi }_i$, for iid noise$$\xi _i$$ ${\xi }_i$, and using the geometric random graph, we identify (with high probability) the rate of convergence of$$u_n$$ $u_n$togin the large data limit$$n\rightarrow \infty $$ $n\to \infty$. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model. 

   more » « less