Lagrangian Coordinates Research Articles

Global weak solutions and long time behavior for 1D compressible MHD equations without resistivity

We study the initial-boundary value problem for 1D compressible magnetohydrodynamics equations of viscous nonresistive fluids in the Lagrangian mass coordinates. Based on the estimates of upper and lower bounds of the density, weak solutions are constructed by approximation of global regular solutions, the existence of which has recently been obtained by Jiang and Zhang [Nonlinearity 30, 3587–3612 (2017)]. The uniqueness of weak solutions is also proved as a consequence of Lipschitz continuous dependence on the initial data. Furthermore, long time behavior for global solutions is investigated. Specifically, based on the uniform-in-time bounds of the density from above and below away from zero, together with the structure of the equations, we show the exponential decay rate in L2- and H1-norm, respectively, with large initial data.

A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary

We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known as co-moving) coordinates, of local-in-time a priori estimates for the solution. The solution features a fluid-vacuum boundary, transported by the fluid four-velocity, along which the hyperbolicity of the equations degenerates. In this context, the relativistic Euler equations are equivalent to a degenerate quasilinear hyperbolic wave-map-like system that cannot be treated using standard energy methods.

Continuum car-following model of capacity drop at sag and tunnel bottlenecks

Abstract Sags and tunnels are major bottlenecks, where the road capacity is reduced, and the “capacity drop” phenomenon occurs; however, there is no simple model or theory that can explain the formation and other characteristics of capacity drop. This paper presents a car-following model, which is equivalent to a continuum model in the Lagrangian coordinates. The model is built on two main assumptions: (i) inhomogeneous fundamental diagrams with location-dependent time gaps, and (ii) bounded acceleration. We first demonstrate that the stationary speed profiles, the low acceleration rates, the dropped capacity, and the approximate time duration of the capacity drop formation in the model are consistent with empirical observations. Then the impacts on the stationary states and dropped capacity of the numerical viscosity caused by the discretization method are investigated, and it is shown that the dropped capacity converges to the theoretical value. Further, a one-dimensional iterated function system is proposed to model the formation mechanism of the capacity drop, which is derived by investigating the spatial pattern of equilibrium and bounded acceleration traffic states that arises in a lead-vehicle problem. Utilizing this model, we uncover a set of properties of the capacity drop such as existence, uniqueness, global convergence, and convergence speed. Finally, the model is applied to analyze the impacts of heterogeneous drivers. The model and insights in this study will help to develop control and management schemes to alleviate capacity drop effects with connected and autonomous vehicles in the future.

Numerical models for evolution of extreme wave groups

The paper considers the application of two numerical models to simulate the evolution of steep breaking waves. The first one is a Lagrangian wave model based on equations of motion of an inviscid fluid in Lagrangian coordinates. A method for treating spilling breaking is introduced and includes dissipative suppression of the breaker and correction of crest shape to improve the post breaking behaviour. The model is used to create a Lagrangian numerical wave tank, to reproduce experimental results of wave group evolution. The same set of experiments is modelled using a novel VoF numerical wave tank created using OpenFOAM. Lagrangian numerical results are validated against experiments and VoF computations and good agreement is demonstrated. Differences are observed only for a small region around the breaking crest.

A continuum model for cities based on the macroscopic fundamental diagram: A semi-Lagrangian solution method

This paper presents a formulation of the reactive dynamic user equilibrium problem in continuum form using a network-level Macroscopic Fundamental Diagram (MFD). Compared to existing continuum models for cities -- all based in Hughes' pedestrian model in 2002 -- the proposed formulation (i) is consistent with reservoir-type models of the MFD literature, shedding some light into the connection between these two modeling approaches, (ii) can have destinations continuously distributed on the region, and (iii) can incorporate multi-commodity flows without additional numerical error. The proposed multi-reservoir numerical solution method treats the multi-commodity component of the model in Lagrangian coordinates, which is the natural representation to propagate origin-destination information (and any vehicle-specific characteristic) through the traffic stream. Fluxes between reservoir boundaries are computed in the Eulerian representation, and are used to calculate the speed of vehicles crossing the boundary. Simple examples are included that show the convergence of the model and its agreements with the available analytical solutions. We find that (i) when origins and destinations are uniformly distributed in a region, the distribution of the travel times can be approximated analytically, (ii) the magnitude of the detours from the optimal free-flow route due to congestion increase linearly with the inflow and decreases with the square of the speed, and (iii) the total delay of vehicles in the network converges to the analytical approximation when the size of reservoirs tends to zero.

Large fluctuations of a Kardar-Parisi-Zhang interface on a half line: The height statistics at a shifted point.

We consider a stochastic interface h(x,t), described by the 1+1 Kardar-Parisi-Zhang (KPZ) equation on the half line x≥0 with the reflecting boundary at x=0. The interface is initially flat, h(x,t=0)=0. We focus on the short-time probability distribution P(H,L,t) of the height H of the interface at point x=L. Using the optimal fluctuation method, we determine the (Gaussian) body of the distribution and the strongly asymmetric non-Gaussian tails. We find that the slower-decaying tail scales as -sqrt[t]lnP≃|H|^{3/2}f_{-}(L/sqrt[|H|t]) and calculate the function f_{-} analytically. Remarkably, this tail exhibits a first-order dynamical phase transition at a critical value of L, L_{c}=0.60223⋯sqrt[|H|t]. The transition results from a competition between two different fluctuation paths of the system. The faster-decaying tail scales as -sqrt[t]lnP≃|H|^{5/2}f_{+}(L/sqrt[|H|t]). We evaluate the function f_{+} using a specially developed numerical method which involves solving a nonlinear second-order elliptic equation in Lagrangian coordinates. The faster-decaying tail also involves a sharp transition which occurs at a critical value L_{c}≃2sqrt[2|H|t]/π. This transition is similar to the one recently found for the KPZ equation on a ring, and we believe that it has the same fractional order, 5/2. It is smoothed, however, by small diffusion effects.

Instability solutions for the Rayleigh–Taylor problem of non-homogeneous viscoelastic fluids in bounded domains

It is well-known that the Rayleigh–Taylor (RT) problem of an inhomogeneous viscoelastic fluid defined on a bounded domain is unstable, if the elasticity coefficient κ is less than some threshold κ C . In this paper, we rigorously prove the existence of a unique unstable strong solution in the sense of L 1 -norm for the RT problem in Lagrangian coordinates based on a bootstrap instability method, when κ < κ C . Applying an inverse transformation of Lagrangian coordinates to the obtained unstable solution, we can further get a unique unstable solution for the RT problem of inhomogeneous viscoelastic fluids in Eulerian coordinates.

Global well‐posedness for the compressible magnetohydrodynamic system in the critical Lp framework

The present work is dedicated to the well‐posedness issue of strong solutions (away from vacuum) to the compressible viscous magnetohydrodynamic (MHD) system in (d ≥ 2). We aim at extending those results in previous studies to more general Lp critical framework. Precisely, by recasting the whole system in Lagrangian coordinates, we prove the local existence and uniqueness of solutions by means of Banach fixed‐point theorem. Furthermore, with the aid of effective velocity, we employ the energy argument to establish global a priori estimates, which lead to the unique global solution near constant equilibrium. Our results hold in case of small data but large highly oscillating initial velocity and magnetic field.

Large Oceanic Gyres: Lagrangian Description

A hydrodynamic model of an oceanic gyre is proposed. The fluid motion is considered in the leading-order shallow-water approximation in the spherical Lagrangian coordinates. Motion of liquid particles at the spherical surfaces is studied versus latitude and longitude as unknown variables. The boundary condition at the edge of the gyre is not formulated. An approximation of the “averaged latitude” is introduced when the coefficients of the momentum equation are replaced by constant values corresponding to the latitude of the gyre’s center. It is shown that the resulting set of equations is similar to the equations of plane hydrodynamics. Its analytical solutions containing two arbitrary functions and two arbitrary constants (time frequencies) are obtained. The trajectories of liquid particles represent a superposition of two rotational motions, and their general properties are discussed. A family of the gyres with invariable shape in time is selected. Their outer boundaries either remain motionless or rotate uniformly. An example of the unsteady gyre both rotating and deforming in its shape is studied numerically.

Fast magnetic reconnection and the ideal evolution of a magnetic field

Regardless of how small non-ideal effects may be, phenomena associated with changes in magnetic field line connections are frequently observed to occur on an Alfvénic time scale. Since it is mathematically impossible for magnetic field line connections to change when non-ideal effects are identically zero, an ideal evolution must naturally lead to states of unbound sensitivity to non-ideal effects. That such an evolution is natural is demonstrated by the use of Lagrangian coordinates based on the flow velocity of the magnetic field lines. The Lagrangian representation of an evolving magnetic field is highly constrained when neither the magnetic field strength nor the forces exerted by the magnetic field increase exponentially with time. The development of a state of fast reconnection consistent with these constraints (1) requires a three-dimensional evolution, (2) has an exponentially increasing sensitivity to non-ideal effects, and (3) has a parallel current density, which lies in exponentially thinning but exponentially widening ribbons, with a magnitude that is limited to a slow growth. The implication is that exponential growth in sensitivity is the cause of fast magnetic reconnection when non-ideal effects are sufficiently small. The growth of the non-ideal effect of the resistivity multiplied by the parallel current density is far too slow to be competitive.

Conservation laws of the two-dimensional gas dynamics equations

Two-dimensional gas dynamics equations in mass Lagrangian coordinates are studied in this paper. The equations describing these flows are reduced to two Euler-Lagrange equations. Using group classification and Noether's theorem, conservation laws are obtained. Their counterparts in Eulerian coordinates are given. Among these counterparts there are new conservation laws.

One-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates: Symmetry classification, conservation laws, difference schemes

Lie point symmetries of the one-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates are considered. Complete Lie group classification of these equations reduced to a scalar second-order PDE is performed. The classification parameter is the entropy. Noether theorem is applied for constructing conservation laws. The conservation laws can be represented in the gas dynamics variables. For the basic adiabatic case invariant and conservative difference schemes are discussed.

On Magnetic Inhibition Theory in Non-resistive Magnetohydrodynamic Fluids

We investigate why the non-slip boundary condition for the velocity, imposed in the direction of impressed magnetic fields, can contribute to the magnetic inhibition effect based on the nonhomogeneous magnetic Rayleigh--Taylor (abbr. NMRT) problem in non-resistive magnetohydrodynamic (abbr. MHD) fluids. Exploiting an infinitesimal method in Lagrangian coordinates, the idea of (equivalent) magnetic tension, and the differential version of magnetic flux conservation, we give an explanation of physical mechanism for the magnetic inhibition phenomenon in a non-resistive MHD fluid. Moreover, we find that the magnetic energy in the non-resistive MHD fluid depends on the displacement of fluid particles, and thus can be regarded as elastic potential energy. Motivated by this observation, we further use the well-known minimum potential energy principle to explain the physical meaning of the stability/instability criteria in the NMRT problem. As a result of the analysis, we further extend the results on the NMRT problem to the stratified MHD fluid case. We point out that our magnetic inhibition theory can be used to explain the inhibition phenomenon of other instabilities, such as thermal instability, magnetic buoyancy instability, and so on, by impressed magnetic fields in non-resistive MHD fluids.

Low Regularity Solutions for Gravity Water Waves

We prove local well-posedness for the gravity water waves equations without surface tension, with initial velocity field in $H^s$, $s > \frac{d}{2} + 1 - \mu$, where $\mu = \frac{1}{10}$ in the case $d = 1$ and $\mu = \frac{1}{5}$ in the case $d \geq 2$, extending previous results of Alazard-Burq-Zuily. The improvement primarily arises in two areas. First, we perform an improved analysis of the regularity of the change of variables from Eulerian to Lagrangian coordinates. Second, we perform a time-interval length optimization of the localized Strichartz estimates.

Multi-body Spherically Symmetric Steady States of Newtonian Self-Gravitating Elastic Matter

We study the problem of static, spherically symmetric, self-gravitating elastic matter distributions in Newtonian gravity. To this purpose we first introduce a new definition of homogeneous, spherically symmetric (hyper)elastic body in Euler coordinates, i.e., in terms of matter fields defined on the current physical state of the body. We show that our definition is equivalent to the classical one existing in the literature and which is given in Lagrangian coordinates, i.e., in terms of the deformation of the body from a given reference state. After a number of well-known examples of constitutive functions of elastic bodies are re-defined in our new formulation, a detailed study of the Seth model is presented. For this type of material the existence of single and multi-body solutions is established.

A transport model for broadening of a linearly polarized, coherent beam due to inhomogeneities in a turbulent atmosphere

ABSTRACTTraditional models for beam broadening include both a diffractive term, owing to the source aperture, and a beam ‘wandering’ term that stems from refractive index variations in the atmosphere. Here we derive a novel beam broadening term that depends on the properties of atmospheric turbulence. The derivation rests on a transport formulation of the propagation problem whereby the magnitude of the electric field is viewed as the density of a fluid, moving in a flow that is driven by the refractive index perturbations. Properties of the transport solutions are obtained using Lagrangian coordinates and are demonstrated to be entirely consistent with existing theory on the subject. The new factor predicts appreciable (25% in our example) increases in beam broadening for applications requiring propagation over very long optical paths and heavy turbulence.

Global strong solutions to 1-D vacuum free boundary problem for compressible Navier–Stokes equations with variable viscosity and thermal conductivity

In this paper, we investigate the vacuum free boundary problem of one-dimensional heat-conducting compressible Navier–Stokes equations where the viscosity coefficient depends on the density, and the heat conductivity coefficient depends on the temperature, satisfying a physical assumption from the Chapman–Enskog expansion of the Boltzmann equation. The fluid connects to the vacuum continuously, thus the system is degenerate near the free boundary. The global existence and uniqueness of strong solutions for the free boundary problem are established when the initial data are large. The result is proved by using both the Lagrangian mass coordinate and the Lagrangian trajectory coordinate. An key observation is that the Jacobian between these coordinates are bounded from above and below by positive constants.

KINEMATICS RENORMALIZATION OF ENERGY IN THE THEORY OF GRAVITY

In this paper, we study the change in the energy-momentum of a gravitational field when the general reference frame is changed. From a mathematical point of view, the general reference frame is described by the affine reference field on the space-time manifold, and the transitions between them form the gauge general linear group . Under the transformations of the group the Lagrangian of the theory, obtains a divergent term: its canonical transformation takes place. Consequently, the gravitational equations do not change, however, the total energy of the system and its superpotential are changes. This phenomenon is called the kinematic renormalization of energy and can be used to deprive the theory of singularities. The paper also examines individual cases of the selection of reference frames, in particular the so-called holonomic reference frames used in the general theory of relativity, as well as non-holonomic pseudoorthonormal reference frames. In the case of holonomic reference frames, it is possible to associate with them coordinate systems whose coordinate rays will set the reference frames. In contrast to the Euler coordinates , which number the points of space-time and whose change does not lead to a change in any physical quantities, the Lagrangian coordinates carry the physical content, since their change leads to a change in the physical reference frame, as a result of which certain physical quantities change, such as the energy-momentum of the system, and so on. In the case of coordinate transformations there are gauge translations in space-time in a fixed general reference frame belonging to the group . In the case of Lagrange coordinates transformations the transformations of holonomic reference frames occur, and such transformations belong to another group, namely the group , forming its subgroup, which is isomorphic to the group of diffeomorphisms due to restriction by holonomic reference frames. In the case of the identification of the Lagrangian coordinates with the Eulerian , the energy-momentum tensor becomes Einstein energy-momentum complex of gravitational field, and its superpotential becomes Freud superpotential. Obtained results open up wide opportunities for depriving the theory of gravity from singularities by choosing the appropriate general reference frames.

Between homogeneous and inhomogeneous Navier–Stokes systems: The issue of stability

We construct large velocity vector solutions to the three dimensional inhomogeneous Navier-Stokes system. The result is proved via the stability of two dimensional solutions with constant density, under the assumption that initial density is point-wisely close to a constant. Key elements of our approach are estimates in the maximal regularity regime and the Lagrangian coordinates. Considerations are done in the whole $\R^3$.

A multicomponent flow model in deformable porous media

We propose a model for multicomponent flow of immiscible fluids in a deformable porous medium accounting for capillary hysteresis. Oil, water, and air in the soil pores offer a typical example of a real situation occurring in practice. We state the problem within the formalism of continuum mechanics as a slow diffusion process in Lagrange coordinates. The balance laws for volumes, masses, and momentum lead to a degenerate parabolic PDE system. In the special case of a rigid solid matrix material and three fluid components, we prove under further technical assumptions that the system is mathematically well posed in a small neighborhood of an equilibrium.