arXiv subjects

Thinh Kieu

Publications and source records attributed to Thinh Kieu.

16 recordsLinked to original sources

arXiv2025

On compressible fluid flows of Forchheimer-type in rotating heterogeneous porous media

We study the dynamics of compressible fluids in rotating heterogeneous porous media. The fluid flow is of {F}orchheimer-type and is subject to a mixed mass and volumetric flux boundary condition. The governing equations are reduced to a nonlinear partial differential equation for the pseudo-pressure. This parabolic-typed equation can be degenerate and/or singular in the spatial variables, the unknown and its gradient. We establish the $L^\alpha$-estimate for the solutions, for any positive number $\alpha$, in terms of the initial and boundary data and the angular speed of rotation. It requires new elliptic and parabolic Sobolev inequalities and trace theorem with multiple weights that are suitable to the nonlinear structure of the equation. The $L^\infty$-estimate is then obtained without imposing any conditions on the $L^\infty$-norms of the weights and the initial and boundary data.

math.AP↗

arXiv2025

A priori estimates for gaseous flows of Forchheimer-type in heterogeneous porous media

We study isentropic fluid flows of gases of the Forchheimer-type in heterogeneous porous media. The governing equation is a doubly nonlinear parabolic equation with coefficients depending on the spatial variables. Its solutions are subject to a nonlinear Robin boundary condition. We establish the estimates of the solutions for short time in terms of the initial and boundary data. For the proof, the multi-weight versions of the Sobolev inequality, parabolic Sobolev inequality and trace theorem are derived. They are then used to implement the Moser iteration for suitable weighted norms.

math.AP↗

arXiv2024

Anisotropic flows of Forchheimer-type in porous media and their steady states

We study the anisotropic Forchheimer-typed flows for compressible fluids in porous media. The first half of the paper is devoted to understanding the nonlinear structure of the anisotropic momentum equations. Unlike the isotropic flows, the important monotonicity properties are not automatically satisfied in this case. Therefore, various sufficient conditions for them are derived and applied to the experimental data. In the second half of the paper, we prove the existence and uniqueness of the steady state flows subject to a nonhomogeneous Dirichlet boundary condition. It is also established that these steady states, in appropriate functional spaces, have local Hölder continuous dependence on the forcing function and the boundary data.

math.AP↗

arXiv2021

Infinite series asymptotic expansions for decaying solutions of dissipative differential equations with non-smooth nonlinearity

We study the precise asymptotic behavior of a non-trivial solution that converges to zero, as time tends to infinity, of dissipative systems of nonlinear ordinary differential equations. The nonlinear term of the equations may not possess a Taylor series expansion about the origin. This absence technically cripples previous proofs in establishing an asymptotic expansion, as an infinite series, for such a decaying solution. In the current paper, we overcome this limitation and obtain an infinite series asymptotic expansion, as time goes to infinity. This series expansion provides large time approximations for the solution with the errors decaying exponentially at any given rates. The main idea is to shift the center of the Taylor expansions for the nonlinear term to a non-zero point. Such a point turns out to come from the non-trivial asymptotic behavior of the solution, which we prove by a new and simple method. Our result applies to different classes of non-linear equations that have not been dealt with previously.

math.CA↗

arXiv2021

A Mixed Finite Element Approximation for Fluid Flows of Mixed Regimes in Porous Media

In this paper, we consider the complex flows when all three regimes pre-Darcy, Darcy and post-Darcy may be present in different portions of a same domain. We unify all three flow regimes under mathematics formulation. We describe the flow of a single-phase fluid in $\R^d, d\ge 2$ by a nonlinear degenerate system of density and momentum. A mixed finite element method is proposed for the approximation of the solution of the above system. The stability of the approximations are proved; the error estimates are derived for the numerical approximations for both continuous and discrete time procedures. The continuous dependence of numerical solutions on physical parameters are demonstrated. Experimental studies are presented regarding convergence rates and showing the dependence of the solution on the physical parameters.

math.NA↗

arXiv2021

Studying a doubly nonlinear model of slightly compressible Forchheimer flows in rotating porous media

We study the generalized Forchheimer flows of slightly compressible fluids in rotating porous media. In the problem's model, the varying density in the Coriolis force is fully accounted for without any simplifications. It results in a doubly nonlinear parabolic equation for the density. We derive a priori estimates for the solutions in terms of the initial, boundary data and physical parameters, emphasizing on the case of unbounded data. Weighted Poincaré-Sobolev inequalities suitable to the equation's nonlinearity, adapted Moser's iteration and maximum principle are used and combined to obtain different types of estimates.

math.AP↗

arXiv2021

Slightly Compressible Forchheimer Flows in Rotating Porous Media

We formulate the the generalized Forchheimer equations for the three-dimensional fluid flows in rotating porous media. By implicitly solving the momentum in terms of the pressure's gradient, we derive a degenerate parabolic equation for the density in the case of slightly compressible fluids and study its corresponding initial, boundary value problem. We investigate the nonlinear structure of the parabolic equation. The maximum principle is proved and used to obtain the maximum estimates for the solution. Various estimates are established for the solution's gradient, in the Lebesgue norms of any order, in terms of the initial and boundary data. All estimates contain explicit dependence on key physical parameters including the angular speed.

math.AP↗

arXiv2019

Solvability of the mixed formulation for generalized Forchheimer flows of isentropic gases

This paper is focused on the generalized Forchheimer flows of isentropic gas, described by a system of two nonlinear degenerating differential equations of first order. We prove the existence and uniqueness of the Dirichlet problem for stationary problem. The technique of semi-discretization in time is used to prove the existence for the time-dependent problem.

math.AP↗

arXiv2018

Existence of a solution for generalized Forchheimer flow in porous media

This paper is focused on the generalized Forchheimer flows for slightly compressible fluids. We prove the existence and uniqueness of the differential system for stationary problem. The technique of semi-discretization in time is used to prove the existence of solution for the transient problem.

math.NA↗

arXiv2016

A mixed finite element approximation for Darcy-Forchheimer flows of slightly compressible fluids

In this paper, we consider the generalized Forchheimer flows for slightly compressible fluids in porous media. Using Muskat's and Ward's general form of Forchheimer equations, we describe the flow of a single-phase fluid in $\mathbb R^d, d\ge 2$ by a nonlinear degenerate system of density and momentum. A mixed finite element method is proposed for the approximation of the solution of the above system. The stability of the approximations are proved; the error estimates are derived for the numerical approximations for both continuous and discrete time procedures. The continuous dependence of numerical solutions on physical parameters are demonstrated. Experimental studies are presented regarding convergence rates and showing the dependence of the solution on the physical parameters.

math.NA↗

arXiv2016

Fluid Flows of Mixed Regimes in Porous Media

In porous media, there are three known regimes of fluid flows, namely, pre-Darcy, Darcy and post-Darcy. Because of their different natures, these are usually treated separately in literature. To study complex flows when all three regimes may be present in different portions of a same domain, we use a single equation of motion to unify them. Several scenarios and models are then considered for slightly compressible fluids. A nonlinear parabolic equation for the pressure is derived, which is degenerate when the pressure gradient is either small or large. We estimate the pressure and its gradient for all time in terms of initial and boundary data. We also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one. Moreover, the continuous dependence of the solutions on initial and boundary data, and the structural stability for the equation are established.

math.AP↗

arXiv2016

Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media

This paper is focused on the generalized Forchheimer flows of compressible fluids in porous media. The gravity effect and other general nonlinear forms of the source terms and boundary fluxes are integrated into the model. It covers isentropic gas flows, ideal gases and slightly compressible fluids. We derive a doubly nonlinear parabolic equation for the so-called pseudo-pressure, and study the corresponding initial boundary value problem. The maximum estimates of the solution are established by using suitable trace theorem and adapting appropriately the Moser's iteration. The gradient estimates are obtained under a theoretical condition which, indeed, is relevant to the fluid flows in applications.

math.AP↗

arXiv2015

Numerical analysis for generalized Forchheimer flows of slightly compressible fluids in porous media

In this paper, we will consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for density. The long time numerical approximation of the nonlinear degenerate parabolic equation with time dependent boundary conditions is studied. The stability for all positive time is established in both a continuous time scheme and a discrete backward Euler scheme. A Gronwall's inequality-type is used to study the asymptotic behavior of the solution. Error estimates for the solution are derived for both continuous and discrete time procedures. Numerical experiments confirm the theoretical analysis regarding convergence rates.

math.NA↗

arXiv2015

Global Estimates for Generalized Forchheimer Flows of Slightly Compressible Fluids

This paper is focused on the generalized Forchheimer flows of slightly compressible fluids in porous media. They are reformulated as a degenerate parabolic equation for the pressure. The initial boundary value problem is studied with time-dependent Dirichlet boundary data. The estimates up to the boundary and for all time are derived for the $L^\infty$-norm of the pressure, its gradient and time derivative. Large-time estimates are established to be independent of the initial data. Particularly, thanks to the special structure of the pressure's nonlinear equation, the global gradient estimates are obtained in a relatively simple way, avoiding complicated calculations and a prior requirement of Hölder estimates.

math.AP↗

arXiv2015

Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluid in porous media

We consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. The prior estimates for the solutions in $L^\infty(0,T;L^q(\Omega)), q\ge 2$, time derivative in $L^\infty(0,T;L^2(\Omega))$ and gradient in $L^\infty(0,T;W^{1,2-a}(\Omega)),$ with $a\in (0,1)$ are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates.

math.NA↗

arXiv2015

Generalized Forchheimer flows of isentropic gases

We consider generalized Forchheimer flows of either isentropic gases or slightly compressible fluids in porous media. By using Muskat's and Ward's general form of the Forchheimer equations, we describe the fluid dynamics by a doubly nonlinear parabolic equation for the appropriately defined pseudo-pressure. The volumetric flux boundary condition is converted to a time-dependent Robin-type boundary condition for this pseudo-pressure. We study the corresponding initial boundary value problem, and estimate the $L^\infty$ and $W^{1,2-a}$ (with $0<a<1$) norms for the solution on the entire domain in terms of the initial and boundary data. It is carried out by using a suitable trace theorem and an appropriate modification of Moser's iteration.

math.AP↗