Relativistic Heat Equation Research Articles

Positive Classical Solutions of Dirichlet Problem for the Steady Relativistic Heat Equation

Positive Classical Solutions of Dirichlet Problem for the Steady Relativistic Heat Equation

On the illposedness and stability of the relativistic heat equation

In this article, we analyze, in terms of a simple example, the incompatibility of parabolic evolution and general covariance. For this, we introduce a unit time-like four-vector and study the simplest heat flux equation with respect to it. In cases where this vector field is surface forming, then the local high wave number limit shows well posedness, but as soon as that property is lost, the Cauchy problem becomes ill-posed. We also discuss how the Maxwell–Cattaneo type modification of the system renders it well posed and link the amplitude of the modification, which is related to the so-called second wave speed of the system, to the size of the failure of surface orthogonality.

The relativistic heat equation on flat Friedman spacetime

The exact solutions of the relativistic hyperbolic heat equation on a flat Friedman spacetime metric are obtained. These solutions are obtained in three cases of flat Friedman model which correspond to radiation, matter and dark energy-dominated universe models. In particular, in the dark energy-dominated universe case, a closed form solution is obtained for which, using appropriate boundary conditions, an interesting exact solution is derived. This solution suggests that the temperature of the universe drops as it expands by a scale of an exponential function.

A convergent Lagrangian discretization for <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Wasserstein and flux-limited diffusion equations

We study a Lagrangian numerical scheme for solving a nonlinear drift diffusion equations of the form $ \partial_t u = \partial_x(u \cdot ({\sf c}^*)^\prime[\partial_x \mathit{h}^\prime(u)+ \mathit{v}^\prime]) $, like Fokker-Plank and $ q $-Laplace equations, on an interval. This scheme will consist of a spatio-temporal discretization founded on the formulation of the equation in terms of inverse distribution functions. It is based on the gradient flow structure of the equation with respect to optimal transport distances for a family of costs that are in some sense $ p $-Wasserstein like. Additionally we will show that, under a regularity assumption on the initial data, this also includes a family of discontinuous, flux-limiting cost inducing equations like Rosenau's relativistic heat equation. We show that this discretization inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, a minimum/maximum principle and flux-limitation in the case of the corresponding cost. Convergence in the limit of vanishing mesh size will be proven as the main result. Finally we will present numerical experiments including a numerical convergence analysis.

Heat conduction in rotating relativistic stars

In the standard form of the relativistic heat equation used in astrophysics, information propagates instantaneously, rather than being limited by the speed of light as demanded by relativity. We show how this equation nonetheless follows from a more general, causal theory of heat propagation in which the entropy plays the role of a fluid. In deriving this result, however, we see that it is necessary to make some assumptions which are not universally valid: the dynamical timescales of the process must be long compared with the explicitly causal physics of the theory, the heat flow must be sufficiently steady, and the spacetime static. Generalising the heat equation (e.g. restoring causality) would thus entail retaining some of the terms we neglected. As a first extension, we derive the heat equation for the spacetime associated with a slowly-rotating star or black hole, showing that it only differs from the static result by an additional advection term due to the rotation, and as a consequence demonstrate that a hotspot on a neutron star will be seen to be modulated at the rotation frequency by a distant observer.

Beyond ideal magnetohydrodynamics: resistive, reactive and relativistic plasmas

We develop a new framework for the modelling of charged fluid dynamics in general relativity. The model, which builds on a recently developed variational multi-fluid framework for dissipative fluids, accounts for relevant effects like the inertia of both charge currents and heat and, for mature systems, the decoupling of superfluid components. We discuss how the model compares to standard relativistic magnetohydronamics and consider the connection between the fluid dynamics, the microphysics and the underlying equation of state. As illustrations of the formalism, we consider three distinct two-fluid models describing (i) an Ohm’s law for resistive charged flows, (ii) a relativistic heat equation, and (iii) an equation representing the momentum of a decoupled superfluid component. As a more complex example, we also formulate a three-fluid model which demonstrates the thermo-electric effect. The new framework allows us to model neutron stars (and related systems) at a hierarchy of increasingly complex levels, and should enable us to make progress on a range of exciting problems in astrophysics and cosmology.

Qualitative behaviour for flux-saturated mechanisms: travelling waves, waiting time and smoothing effects

This paper is devoted to the analysis of qualitative properties of flux-saturated type operators in dimension one. Specifically, we study regularity properties and smoothing effects, discontinuous interfaces, the existence of traveling wave profiles, sub- and super-solutions and waiting time features. The aim of the paper is to better understand these kind of phenomena throughout two prototypic operators: The relativistic heat equation and the porous media flux-limited equation. As an important consequence of our results we deduce that solutions to the one-dimensional relativistic heat equation become smooth inside their support on the long time run.

Analytical and numerical solutions to some one-dimensional relativistic heat equations

Analytical and numerical solutions to a family of one-dimensional (nonlinear) relativistic heat equations in finite domains are presented. The analytical solutions correspond to steady state conditions in the absence of source terms and have been obtained as functions of the (absolute) temperature at one of the boundaries, a characteristic exponent, a Péclet number based on the speed of light and the heat flux, while the numerical ones correspond to an initial Gaussian temperature distribution, adiabatic boundary conditions and different values of the Péclet number and a characteristic exponent. It is shown that, for steady conditions, the difference between the (nondimensional) temperature of the relativistic heat equation and that corresponding to Fourier law is very large for large values of both the coefficient and the exponent of the nonlinearity that characterize the relativistic contribution to the heat flux, small values of the temperature at one of the boundaries and large heat fluxes. Travelling-wave solutions of the wave-front type are reported for odd values of the nonlinearity exponent in infinite domains and in the absence of source terms. For an initial Gaussian distribution, it is shown that the relativistic contribution to heat transfer results in the formation of two triangular corner regions where the temperature is equal to the initial one, and the formation of two temperature fronts that propagate towards the domain’s boundaries. The amplitude and steepness of these fronts increase whereas their width and speed decrease as the Péclet number is decreased. It is also shown that the effects of the characteristic exponent are small provided that its value is greater than about two, and that, in the absence of source terms, the temperature becomes uniform in space and constant in time for adiabatic boundary conditions. In the presence of source terms and for adiabatic boundary conditions, it is shown that, soon after the temperature fronts hit the boundaries, the temperature becomes uniform in space but may either increase or decrease with time until it reaches a stable fixed point of the source term. For a cubic source term that exhibits bistability, it is shown that the temperature tends to the attractor of lowest temperature.

Formulation of the relativistic heat equation and the relativistic kinetic Fokker–Planck equations using GENERIC

In this paper, we formulate the relativistic heat equation and the relativistic kinetic Fokker–Planck equations into the GENERIC (General Equation for Non-Equilibrium Reversible–Irreversible Coupling) framework. We also show that the relativistic Maxwellian distribution is the stationary solution of the latter. The GENERIC formulation provides an alternative justification that the two equations are meaningful relativistic generalizations of their non-relativistic counterparts.

Erratum to: The Dirichlet problem associated to the relativistic heat equation

In the original article we proved existence and uniqueness of entropy solutions for the nonhomogeneous Dirichlet problem associated to the relativistic heat equation. In the proof of uniqueness of both elliptic and parabolic entropy solutions (Theorems 2 and 3, respectively) we used that sign0 (u− u)u ∈ BV ( ) for two entropy solutions u, u. However, we did not prove this but we wrongly stated that this was true for any given functions u, u ∈ TBV( ), which is not the case, in general. The expression sign0 (u − u)u appears in the process of applying the doubling variables method, known as Kruzhkov’s method, if one lets the parameter e → 0+, which serves to approximate the sign function, before joining the space variables. In this note, we correct the aforementioned mistake by joining first the space variables and at the very end, letting e → 0+.

Fourier, hyperbolic and relativistic heat transfer equations: a comparative analytical study

Parabolic heat equation based on Fourier's theory (FHE), and hyperbolic heat equation (HHE), has been used to mathematically model the temperature distributions of biological tissue during thermal ablation. However, both equations have certain theoretical limitations. The FHE assumes an infinite thermal energy propagation speed, whereas the HHE might possibly be in breach of the second law of thermodynamics. The relativistic heat equation (RHE) is a hyperbolic-like equation, whose theoretical model is based on the theory of relativity and which was designed to overcome these theoretical impediments. In this study, the three heat equations for modelling of thermal ablation of biological tissues (FHE, HHE and RHE) were solved analytically and the temperature distributions compared. We found that RHE temperature values were always lower than those of the FHE, while the HHE values were higher than the FHE, except for the early stages of heating and at points away from the electrode. Although both HHE and RHE are mathematically hyperbolic, peaks were only found in the HHE temperature profiles. The three solutions converged for infinite time or infinite distance from the electrode. The percentage differences between the FHE and the other equations were larger for higher values of thermal relaxation time in HHE.

On the relativistic heat equation in one space dimension

We study the relativistic heat equation in one space dimension. We prove a local regularity result when the initial datum is locally Lipschitz in its support. We propose a numerical scheme that captures the known features of the solutions and allows for analysing further properties of their qualitative behavior.

Fronts propagating with signal dependent speed in limited diffusion and related Hamilton–Jacobi formulations

We consider a class of limited diffusion equations and explore the formation of diffusion fronts as the result of a combination of diffusive and hyperbolic transport. We analyze a new class of Hamilton–Jacobi equations arising from the convective part of general Fokker–Planck equations ruled by a non-negative diffusion coefficient that depends on the unknown and on the gradient of the unknown. We explore the main features of the solution of the Hamilton–Jacobi equations that contain shocks and propose a suitable numerical scheme that approximates the solution in a consistent way with respect to the solution of the associated Fokker–Planck equation. We analyze three model problems covering different scenarios. One is the relativistic heat equation model where the speed of propagation of fronts is constant. A second one is a standard porous media model where the speed of propagation of fronts is a function of the density, is unbounded and can exceed any fixed value. We propose a third one which is a porous media model whose speed of propagating fronts depends on the density media and is limited. The three model problems satisfy a general Darcy law. We perform a set of numerical experiments under different piecewise smooth initial data with compact support and compare the behavior of the three different model problems.

Diffusion in curved spacetimes

Using simple kinematical arguments, we derive the Fokker–Planck equation for diffusion processes in curved spacetimes. In the case of pure Brownian motion, this equation coincides with Eckart's relativistic heat equation (albeit in a simpler form) and therefore provides a microscopic justification of his phenomenological heat-flux ansatz. Furthermore, it is easy to derive from it the small-time asymptotic expansion of the mean square displacement of Brownian motion in static spacetimes. Beyond general relativity itself, this result has potential applications in analogue gravitational systems.

The optimal mass transport problem for relativistic costs

In this paper, we study the optimal mass transportation problem in \({\mathbb{R}^{d}}\) for a class of cost functions that we call relativistic cost functions. Consider as a typical example, the cost function c(x, y) = h(x − y) being the restriction of a strictly convex and differentiable function to a ball and infinite outside this ball. We show the existence and uniqueness of the optimal map given a relativistic cost function and two measures with compact support, one of the two being absolutely continuous with respect to the Lebesgue measure. With an additional assumption on the support of the initial measure and for supercritical speed of propagation, we also prove the existence of a Kantorovich potential and study the regularity of this map. Besides these general results, a particular attention is given to a specific cost because of its connections with a relativistic heat equation as pointed out by Brenier (Extended Monge–Kantorovich Theory. Optimal Transportation and Applications, 2003).

Radially Symmetric Solutions of a Tempered Diffusion Equation. A Porous Media, Flux-Limited Case

In this paper we provide analytical and/or numerical evidences of the qualitative properties of discontinuity fronts, developed by the solutions of the relativistic heat equation (and some porous media variants). We study the local-in-time existence of radially symmetric smooth solutions (inside the support) for smooth initial conditions whose only discontinuities are at the boundary of its support. Then we show some numerical experiments that permit us to conjecture the regularity and qualitative properties of entropy solutions of the relativistic heat equation and some porous media–type variants.

On the entropy conditions for some flux limited diffusion equations

In this paper we give a characterization of the notion of entropy solutions of some flux limited diffusion equations for which we can prove that the solution is a function of bounded variation in space and time. This includes the case of the so-called relativistic heat equation and some generalizations. For them we prove that the jump set consists of fronts that propagate at the speed given by Rankine–Hugoniot condition and we give on it a geometric characterization of the entropy conditions. Since entropy solutions are functions of bounded variation in space once the initial condition is, to complete the program we study the time regularity of solutions of the relativistic heat equation under some conditions on the initial datum. An analogous result holds for some other related equations without additional assumptions on the initial condition.

A View on Optimal Transport from Noncommutative Geometry

We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. Starting from a remark of Rieffel on compact manifolds, we first show that on any - i.e. non-necessary compact - complete Riemannian spin manifolds, the two distances coincide. Then, on convex manifolds in the sense of Nash embedding, we provide some natural upper and lower bounds to the distance between any two probability distributions. Specializing to the Euclidean space $R^n$, we explicitly compute the distance for a particular class of distributions generalizing Gaussian wave packet. Finally we explore the analogy between the spectral and the Wasserstein distances in the noncommutative case, focusing on the standard model and the Moyal plane. In particular we point out that in the two-sheet space of the standard model, an optimal-transport interpretation of the metric requires a cost function that does not vanish on the diagonal. The latest is similar to the cost function occurring in the relativistic heat equation.

Diffusion front capturing schemes for a class of Fokker–Planck equations: Application to the relativistic heat equation

In this research work we introduce and analyze an explicit conservative finite difference scheme to approximate the solution of initial-boundary value problems for a class of limited diffusion Fokker–Planck equations under homogeneous Neumann boundary conditions. We show stability and positivity preserving property under a Courant–Friedrichs–Lewy parabolic time step restriction. We focus on the relativistic heat equation as a model problem of the mentioned limited diffusion Fokker–Planck equations. We analyze its dynamics and observe the presence of a singular flux and an implicit combination of nonlinear effects that include anisotropic diffusion and hyperbolic transport. We present numerical approximations of the solution of the relativistic heat equation for a set of examples in one and two dimensions including continuous initial data that develops jump discontinuities in finite time. We perform the numerical experiments through a class of explicit high order accurate conservative and stable numerical schemes and a semi-implicit nonlinear Crank–Nicolson type scheme.

The Dirichlet problem associated to the relativistic heat equation

We prove existence and uniqueness of entropy solutions for the nonhomogeneous Dirichlet problem associated to the relativistic heat equation.