Lagrangian Coordinates Research Articles

Relaxation to Equilibrium in the One-Dimensional Thin-Film Equation with Partial Wetting and Linear Mobility

We investigate the large time behavior of compactly supported smooth solutions for a one-dimensional thin-film equation with linear mobility in the regime of partial wetting. We show the stability of steady state solutions. Relaxation rates are obtained for initial data which are close to a steady state in a suitable sense. The proof uses the Lagrangian coordinates. Our method is to establish and exploit differential relations between the energy and the dissipation as well as some interpolation inequalities. Our result is different from earlier results because here we consider solutions with finite mass.

Description of the Expansion of a Two-Layer Tube: An Analytic Plane-Strain Solution for Arbitrary Pressure-Independent Yield Criterion and Hardening Law

The main objective of the present paper is to provide a simple analytical solution for describing the expansion of a two-layer tube under plane-strain conditions for its subsequent use in the preliminary design of hydroforming processes. Each layer’s constitutive equations are an arbitrary pressure-independent yield criterion, its associated plastic flow rule, and an arbitrary hardening law. The elastic portion of strain is neglected. The method of solution is based on two transformations of space variables. Firstly, a Lagrangian coordinate is introduced instead of the Eulerian radial coordinate. Then, the Lagrangian coordinate is replaced with the equivalent strain. The solution reduces to ordinary integrals that, in general, should be evaluated numerically. However, for two hardening laws of practical importance, these integrals are expressed in terms of special functions. Three geometric parameters for the initial configuration, a constitutive parameter, and two arbitrary functions classify the boundary value problem. Therefore, a detailed parametric analysis of the solution is not feasible. The illustrative example demonstrates the effect of the outer layer’s thickness on the pressure applied to the inner radius of the tube.

Singularities and unsteady separation for the inviscid two-dimensional Prandtl system

We consider the inviscid unsteady Prandtl system in two dimensions, motivated by the fact that it should model to leading order separation and singularity formation for the original viscous system. We give a sharp expression for the maximal time of existence of regular solutions, showing that singularities only happen at the boundary or on the set of zero vorticity, and that they correspond to boundary layer separation. We then exhibit new Lagrangian formulae for backward self-similar profiles, and study them also with a different approach that was initiated by Elliott–Smith–Cowley and Cassel–Smith–Walker. One particular profile is at the heart of the so-called Van-Dommelen and Shen singularity, and we prove its generic appearance (that is, for an open and dense set of blow-up solutions) for any prescribed Eulerian outer flow. We comment on the connection between these results and the full viscous Prandtl system. This paper combines ideas for transport equations, such as Lagrangian coordinates and incompressibility, and for singularity formation, such as self-similarity and renormalisation, in a novel manner, and designs a new way to study singularities for quasilinear transport equations.

Magnetic reconnection and thermal equilibration

When a magnetic field is forced to evolve on a time scale τev, as by footpoint motions driving the solar corona or non-axisymmetric instabilities in tokamaks, the magnetic field lines undergo large-scale changes in topology on a time scale approximately an order of magnitude longer than τev. But the physics that allows such changes operates on a time scale eight or more orders of magnitude slower. An analogous phenomenon occurs in air. Temperature equilibration occurs on a time scale approximately an order of magnitude longer than it takes air to cross a room, τev, although the physical mechanism that allows temperature equilibration is approximately four orders of magnitude slower than τev. The use of Lagrangian coordinates allows the fundamental equations to be solved and both phenomena explained. The theories of thermal equilibration and magnetic reconnection are developed in parallel to help readers obtain an understanding of the importance and implications of analyses using Lagrangian coordinates.

A finite strain solution for strain-softening rock mass around circular roadways

This paper presents a semi-analytical finite strain solution for circular roadways excavated in a strain-softening rock mass with a non-associated flow rule. The evolutions of both strength parameters and Young’s modulus are considered, and the finite strains are applied to describe both the elastic and plastic behaviors with the logarithmic strain in Lagrangian coordinates. By transforming the derivative of displacement with a new variable that is correlated with displacement, the stress and displacement in the elastic region are obtained in the form of integral equations. Considering the assumption of constant material parameters within a very small region, the plastic region is divided into a number of concentric annuli, whose closed-form solutions of displacement and stress can be recursively obtained. The solution is validated by similarity solutions and numerical simulation. Both the roadway wall displacement and thickness of the plastic (residual) region appear to be smaller than those of the small strain solution. For the elasto-perfectly-plastic rock mass, the finite strain solution is approximately the same as that of small strain solution. However, for the strain-softening rock mass, with the decrease of Young’s modulus, the thickness of plastic and residual regions shows the first increase and then decrease trend, finally approaching nil; the difference between the thickness of softening and residual region gradually decreases. The decreasing Young’s modulus deterioration coefficient and increasing dilation angle significantly produce the displacement. Both the displacement and the thickness of plastic and residual regions decrease with the increase of Poisson’s ratio. When Young’s modulus is large, the tangential stress gradually decreases with the increasing radius. When Young’s modulus is small, the tangential stress firstly increases to a maximum value, then decreases gradually. Different from the small strain solutions, the roadway wall displacement of finite strain solutions cannot be larger than the excavation radius, which is in compliance with the practical underground engineering condition. The predicted displacement and thickness in both plastic and residual regions are in accordance with the field test results.

Strong solutions of the equations for viscoelastic fluids in some classes of large data

We study the existence and uniqueness of global strong solutions to the equations of an incompressible viscoelastic fluid in a spatially periodic domain, and show that a unique strong solution exists globally in time if the initial deformation and velocity are small for the given physical parameters. In particular, the initial velocity can be large for the large elasticity coefficient. Our result mathematically verifies that the elasticity can prevent the formation of singularities of strong solutions with large initial velocity, thus playing a similar role to viscosity in preventing the formation of singularities in viscous flows. Moreover, for given initial velocity perturbation and zero initial deformation around the rest state, we find, as the elasticity coefficient or time goes to infinity, that(1)any straight line segment l0 consisted of fluid particles in the rest state, after being bent by a velocity perturbation, will turn into a straight line segment that is parallel to l0 and has the same length as l0.(2)the motion of the viscoelastic fluid can be approximated by a linear pressureless motion in Lagrangian coordinates, even when the initial velocity is large. Moreover, the above mentioned phenomena can also be found in the corresponding compressible fluid case.

A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa–Holm system

We define a kinetic and a potential energy such that the principle of stationary action from Lagrangian mechanics yields a Camassa–Holm system (2CH) as the governing equations. After discretizing these energies, we use the same variational principle to derive a semi-discrete system of equations as an approximation of the 2CH system. The discretization is only available in Lagrangian coordinates and requires the inversion of a discrete Sturm–Liouville operator with time-varying coefficients. We show the existence of fundamental solutions for this operator at initial time with appropriate decay. By propagating the fundamental solutions in time, we define an equivalent semi-discrete system for which we prove that there exists unique global solutions. Finally, we show how the solutions of the semi-discrete system can be used to construct a sequence of functions converging to the conservative solution of the 2CH system.

Multiclass Traffic Flow Dynamics: An Endogenous Model

Vehicles on roads can be distinguished, each defined by its own set of properties (e.g., fleet length and free-flow speed). The traffic states on roads can be attributed to the longitudinal heterogeneity in vehicles. Vehicles slower than prevailing vehicles are defined as moving bottlenecks. On a multilane road section with multiple vehicle types, slower vehicles create moving bottlenecks and induce overtaking by faster vehicles so as to maintain their higher desired speed. The influence of single-class moving bottlenecks has been studied in the past. However, the impacts of multiple classes of moving bottlenecks have not yet been fully explored. This paper categorizes vehicles into passenger cars, medium trucks, and heavy trucks. By defining medium trucks and heavy trucks as moving bottlenecks, we develop analytical formulas for the fundamental diagram on a multilane road section with heterogeneous moving bottlenecks. The formula confirms that the composition of traffic and the longest truck platoon length influence the fundamental diagram. We then conduct simulations using a first-order kinematic wave model in Lagrangian coordinates to validate the fundamental diagram developed with the analytical formula and obtain promising results. This study provides fundamental knowledge for multiclass traffic modeling and multilane traffic operations.

Finite Pure Plane Strain Bending of Inhomogeneous Anisotropic Sheets

The present paper concerns the general solution for finite plane strain pure bending of incompressible, orthotropic sheets. In contrast to available solutions, the new solution is valid for inhomogeneous distributions of plastic properties. The solution is semi-analytic. A numerical treatment is only necessary for solving transcendent equations and evaluating ordinary integrals. The solution’s starting point is a transformation between Eulerian and Lagrangian coordinates that is valid for a wide class of constitutive equations. The symmetric distribution relative to the center line of the sheet is separately treated where it is advantageous. It is shown that this type of symmetry simplifies the solution. Hill’s quadratic yield criterion is adopted. Both elastic/plastic and rigid/plastic solutions are derived. Elastic unloading is also considered, and it is shown that reverse plastic yielding occurs at a relatively large inside radius. An illustrative example uses real experimental data. The distribution of plastic properties is symmetric in this example. It is shown that the difference between the elastic/plastic and rigid/plastic solutions is negligible, except at the very beginning of the process. However, the rigid/plastic solution is much simpler and, therefore, can be recommended for practical use at large strains, including calculating the residual stresses.

Stability of smooth solutions for the compressible Euler equations with time-dependent damping and one-side physical vacuum

In this paper, the one-side physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the physical vacuum boundary, the sound speed is C1/2-Hölder continuous. The coefficient of the time-dependent damping is given by μ(1+t)λ, (0<λ,0<μ) which decays by order −λ in time. First we give an one-side physical vacuum background solution whose density and velocity have a growing order with respect to time. Then the main purpose of this paper is to prove the stability of this background solution under the assumption that 0<λ<1,0<μ or λ=1,2<μ. The pointwise convergence rate of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on the space-time weighted energy estimates, elliptic estimates and the Hardy inequality in the Lagrangian coordinates.

КОНЕЧНЫЕ ДЕФОРМАЦИИ ТОРОИДАЛЬНОЙ ОБОЛОЧКИ

The stress-strain state of a nonlinear elastic shell exposed to the internal pressure is considered. A surface of the shell is toroidal in shape in the initial state. The Lagrangian coordinates of the shell are assigned to a cylindrical system. The kinematic characteristics of the process are shown: a law of the motion of points, vectors of a material basis, a strain affinor and its polar decomposition, the Cauchy-Green strain measure and tensor, the Finger measure, and the “left” and the“right” Hencky strain tensors. Neglecting the shear components of the stress tensor, a constitutive relation is obtained as a quasilinear relation between true stress tensor and the Hencky corotation tensor. A system of equilibrium equations is presented in terms of physical components of the true stress tensor in the Lagrangian coordinates. Using the equilibrium equations and the incompressibility condition, a closed system of nonlinear ordinary differential equations is obtained to determine six unknown functions, depending on the angle indicating a position of the points along the cross-section in the initial state. The method of successive approximations is applied to estimate stress tensor components and to derive logarithms of the elongations of material fibers.

A priori Estimates for the Incompressible Free-Boundary Magnetohydrodynamics Equations with Surface Tension

We consider the three-dimensional incompressible free-boundary magnetohydrodynamics (MHD) equations in a bounded domain with surface tension on the boundary. We establish a priori estimate for solutions in the Lagrangian coordinates with $H^{3.5}$ regularity. To the best of our knowledge, this is the first result focusing on the incompressible ideal free-boundary MHD equations with surface tension. It is worth pointing out that the $1/2$-extra spatial regularity for the flow map $\eta$ is no longer required in this manuscript thanks to the presence of the surface tension on the boundary.

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ КОНТАКТНОЇ ВЗАЄМОДІЇ ПОПЕРЕДНЬО НАПРУЖЕНОГО КІЛЬЦЕВОГО ШТАМПА ТА ПРУЖНОГО ПІВПРОСТОРУ З ПОЧАТКОВИМИ НАПРУЖЕНЯМИ

Стаття присвячена математичному моделюванню контактної взаємодії попередньо напруженого кільцевого штампа та пружного півпростору з початковими (залишковими) напруженнями. Представлено розв’язок для кільцевого пружного штампа, що враховує вплив початкових напружень. Задачу розв’язано у випадку рівних коренів визначального рівняння. Дослідження представлено в загальному вигляді для теорії великих початкових деформацій і двох варіантів теорії малих початкових деформацій у рамках лінеаризованої теорії пружності при довільній структурі пружного потенціалу. Припускається, що початкові стани пружного кільцевого штампа та пружного півпростору однорідні та рівні. Дослідження проводиться в координатах початкового деформованого стану, які пов'язані з лагранжевими координатами (природного стану). Крім того, вплив кільцевого штампа викликає невеликі збурення відповідних величин основного напружено-деформованого стану. Також передбачається, що пружний кільцевий штамп та пружний півпростір виготовлені з різних ізотропних, трансверсально-ізотропних або композиційних матеріалів. Наведені загальні розв’язки основних диференціальних рівнянь лінеаризованої теорії пружності у випадку осесиметричної деформації для скінченної кільцевої області. У результаті, розв’язки поставленої задачі представлені у вигляді нескінченних рядів, коефіцієнти яких визначаються з нескінченної системи алгебраїчних рівнянь. Проведено дослідження впливу початкових (залишкових) напружень у півпросторі та кільцевому штампі на розподіл контактних характеристик в області контакту. У випадку рівних коренів та потенціалу Бартенєва-Хазановича наведено результати чисельного аналізу, що подані у вигляді графіків, які ілюструють достатньо значний вплив початкових напружень. Отже, вплив початкових напружень на напружено-деформований стан пружного кільцевого штампа, що втискається у пружний півпростір, полягає у тому, що: у випадку стиснення початкові напруження в півпросторі призводять до зменшення напружень у пружному штампі, а у випадку розтягу – до їх збільшення, а для переміщень – навпаки.

The Waiting Time Phenomenon in Spatially Discretized Porous Medium and Thin Film Equations

Various degenerate diffusion equations exhibit a waiting time phenomenon: depending on the “flatness” of the compactly supported initial datum at the boundary of the support, the support of the solution may not expand for a certain amount of time. We show that this phenomenon is captured by particular Lagrangian discretizations of the porous medium and the thin film equations, and we obtain sufficient criteria for the occurrence of waiting times that are consistent with the known ones for the original PDEs. For the spatially discrete solution, the waiting time phenomenon refers to a deviation of the edge of support from its original position by a quantity comparable to the mesh width, over a mesh-independent time interval. Our proof is based on estimates on the fluid velocity in Lagrangian coordinates. Combining weighted entropy estimates with an iteration technique à la Stampacchia leads to upper bounds on free boundary propagation. Numerical simulations show that the phenomenon is already clearly visible for relatively coarse discretizations.

On the Cauchy problem for Aw-Rascle system with linear damping

The existence of global BV solutions for the Aw-Rascle system with linear damping is considered. In order to get approximate solutions we consider the system in Lagrangian coordinates, then by using the wave front tracking method coupling with and suitable splitting algorithm and the ideas of [1] we get a sequence of approximate solutions. Finally we show the convergence of this approximate sequence to the weak entropic solution.

A ghost structure finite difference method for a fractional FitzHugh-Nagumo monodomain model on moving irregular domain

In this paper, a ghost structure (GS) finite difference method is proposed to simulate the fractional FitzHugh-Nagumo (FHN) monodomain model on a moving irregular computational domain. In the GS formulation the moving irregular domain is converted into a fixed regular domain (called ghost structure), and the transmembrane potential is described in the Eulerian coordinates, while the membrane dynamics are described in the Lagrangian coordinates. The transformation between the Lagrangian variables and the Eulerian variables is achieved by an integral transformation which involves a delta function. The GS formulation allows to compute the transmembrane potential in a fixed regular domain using the finite difference method on a Cartesian grid, which has a huge advantage for approximating domain-dependent fractional derivatives. To overcome the difficulty caused by running time-consuming loops to compute the transformation between the Eulerian and Lagrangian variables, two fast algorithms are proposed to compute the transformation. Extensive numerical tests are provided to demonstrate the effectiveness and robustness of the proposed GS finite difference method for solving the fractional FHN monodomain model. We first numerically study the transmembrane potential propagation in both healthy hearts and hearts with arrhythmia by simulating the model in the transverse of a ventricle. We then study the transmembrane potential propagation during the pumping process, which requires to simulate the model in the moving longitudinal section of a ventricle. Our numerical results show that the change of spatial derivatives can affect the propagation velocity and the width of the transmembrane potential wave, and for a heart with arrhythmia, the transmembrane potential begins to enter cyclically the region where cardiomyocytes have been excited and then stimulates cardiomyocytes to contract again.

Hydrodynamic limit for a diffusive system with boundary conditions

We study the hydrodynamic limit for the isothermal dynamics of an anharmonic chain under hyperbolic space-time scaling and with nonvanishing viscosity. The temperature is kept constant by a contact with a heat bath, realised via a stochastic momentum-preserving noise added to the dynamics. The noise is designed so it contributes to the macroscopic limit. Dirichlet boundary conditions are also considered: one end of the chain is kept fixed, while a time-varying tension is applied to the other end. Moreover, Neumann boundary conditions are added in such a way that the system produces the correct thermodynamic entropy in the macroscopic limit. We show that the volume stretch and momentum converge (in an appropriate sense) to a smooth solution of a system of parabolic conservation laws (isothermal Navier-Stokes equations in Lagrangian coordinates) with boundary conditions. Finally, changing the external tension allows us to define thermodynamic isothermal transformations between equilibrium states. We use this to deduce the first and the second law of Thermodynamics for our model.

The one-dimensional Green–Naghdi equations with a time dependent bottom topography and their conservation laws

This paper deals with the one-dimensional Green–Naghdi equations describing the behavior of fluid flow over an uneven bottom topography depending on time. Using Matsuno’s approach, the corresponding equations are derived in Eulerian coordinates. Further study is performed in Lagrangian coordinates. This study allowed us to find the general form of the Lagrangian corresponding to the analyzed equations. Then, Noether’s theorem is used to derive conservation laws. As some of the tools in the application of Noether’s theorem are admitted generators, a complete group classification of the Green–Naghdi equations with respect to the bottom depending on time is performed. Using Noether’s theorem, the found Lagrangians, and the group classification, conservation laws of the one-dimensional Green–Naghdi equations with uneven bottom topography depending on time are obtained.

Global existence and convergence to the modified Barenblatt solution for the compressible Euler equations with physical vacuum and time-dependent damping

In this paper, the smooth solution of the physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the vacuum boundary, the sound speed is $C^{1/2}$-H\"{o}lder continuous. The coefficient of the damping depends on time, given by this form $\frac{\mu}{(1+t)^\lambda}$, $\lambda$, $\mu>0$, which decays by order $-\lambda$ in time. Under the assumption that $0<\lambda<1$, $0<\mu$ or $\lambda=1$, $2<\mu$, we will prove the global existence of smooth solutions and convergence to the modified Barenblatt solution of the related porous media equation with time-dependent dissipation and the same total mass when the initial data of the Euler equations is a small perturbation of that of the Barenblatt solution. The pointwise convergence rates of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on space-time weighted energy estimates, elliptic estimates and Hardy inequality in the Lagrangian coordinates. Our result is an extension of that in Luo-Zeng [Comm. Pure Appl. Math. 69 (2016), no. 7, 1354-1396], where the authors considered the physical vacuum free boundary problem of the compressible Euler equations with constant-coefficient damping.

Global well-posedness for the incompressible Navier–Stokes equations in the critical Besov space under the Lagrangian coordinates

We consider global well-posedness of the Cauchy problem of the incompressible Navier–Stokes equations under the Lagrangian coordinates in scaling critical Besov spaces. We prove the system is globally well-posed in the homogeneous Besov space B˙p,1−1+n/p(Rn) with 1≤p<∞. The former result was restricted for 1≤p<2n and the main reason why the well-posedness space is enlarged is that the quasi-linear part of the system has a special feature called a multiple divergence structure and the bilinear estimate for the nonlinear terms are improved by such a structure. Our result indicates that the Navier–Stokes equations can be transferred from the Eulerian coordinates to the Lagrangian coordinates even for the solution in the limiting critical Besov spaces.