Conservation laws of the one-dimensional isentropic gas dynamics equations in Lagrangian coordinates (original) (raw)

Isothermal Limit of Entropy Solutions of the Euler Equations for Isentropic Gas Dynamics

2022

We are concerned with the isothermal limit of entropy solutions in L, containing the vacuum states, of the Euler equations for isentropic gas dynamics. We prove that the entropy solutions in L of the isentropic Euler equations converge strongly to the corresponding entropy solutions of the isothermal Euler equations, when the adiabatic exponent γ → 1. This is achieved by combining careful entropy analysis and refined kinetic formulation with compensated compactness argument to obtain the required uniform estimates for the limit. The entropy analysis involves careful estimates for the relation between the corresponding entropy pairs for the isentropic and isothermal Euler equations when the adiabatic exponent γ → 1. The kinetic formulation for the entropy solutions of the isentropic Euler equations with the uniformly bounded initial data is refined, so that the total variation of the dissipation measures in the formulation is locally uniformly bounded with respect to γ > 1.

Plane one-dimensional MHD flows: symmetries and conservation laws

2021

The paper considers the plane one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. The inviscid, thermally non-conducting medium is modeled by a polytropic gas. The equations are examined for symmetries and conservation laws. For the case of the finite electric conductivity we establish Lie group classification, i.e. we describe all cases of the conductivity σ(ρ, p) for which there are symmetry extensions. The conservation laws are derived by the direct computation. For the case of the infinite electrical conductivity the equations can be brought into a variational form in the Lagrangian coordinates. Lie group classification is performed for the entropy function as an arbitrary element. Using the variational structure, we employ the Noether theorem for obtaining conservation laws. The conservation laws are also given in the physical variables.

Existence Theory for the Isentropic Euler Equations

Archive for Rational Mechanics and Analysis, 2003

We establish an existence theorem for entropy solutions to the Euler equations modeling isentropic compressible fluids. We develop a new approach for constructing mathematical entropies for the Euler equations, which are singular near the vacuum. In particular, we identify the optimal assumption required on the singular behavior on the pressure law at the vacuum in order to validate the two-term asymptotic expansion of the entropy kernel we proposed earlier. For more general pressure laws, we introduce a new multiple-term expansion based on the Bessel functions with suitable exponents, and we also identify the optimal assumption needed to validate the multiple-term expansion and to establish the existence theory. Our results cover, as a special example, the density-pressure law p(ρ) = κ 1 ρ γ 1 + κ 2 ρ γ 2 where γ 1 , γ 2 ∈ (1, 3) and κ 1 , κ 2 > 0 are arbitrary constants.

Nonlinear Conservation Laws and Related Problems

2010

Typical examples of ``nonlinear conservation laws" are the Euler equations, MHD equations, Navier-Stokes equations, Boltzmann equation, and other important models arising in Elasticity, Fluid Dynamics, Combustion, and Kinetic theory. The Euler equations for inviscid compressible fluid flow are fundamental and important to many applications, yet the multidimensional theory is difficult and challenging. Presentation Highlights In this section we present a description of the keynote lectures of the meeting as well as the topics discussed during the panel discussions. The main themes of the keynote lectures are: (a) Singular Limits in Hydrodynamics (b) Recent Developments in Numerical Methods for Nonlinear PDEs (c) Local Isometric Embedding of Surfaces with Nonpositive Gaussian Curvature (d) Kaehler Geometry from PDE Perspective.

A Lagrangian approach for scalar multi-d conservation laws

2017

We introduce a notion of Lagrangian representation for entropy solutions to scalar conservation laws in several space dimension { ∂tu + divx(f(u)) = 0 (t, x) ∈ (0,+∞)× Rd, u(0, x) = u0 t = 0. The construction is based on the transport collapse method introduced by Brenier. As a first application we show that if the solution u is continuous, then it is hypograph is given by the set { (t, x, h) : h ≤ u0(x− f(h)t) } , i.e. it is the translation of each level set of u0 by its characteristic speed. Preprint SISSA 36/2017/MATE

Euler Equations and Related Hyperbolic Conservation Laws

Handbook of Differential Equations Evolutionary Equations, 2005

Some aspects of recent developments in the study of the Euler equations for compressible fluids and related hyperbolic conservation laws are analyzed and surveyed. Basic features and phenomena including convex entropy, symmetrization, hyperbolicity, genuine nonlinearity, singularities, BV bound, concentration and cavitation are exhibited. Global well-posedness for discontinuous solutions, including the BV theory and the L ~ theory, for the one-dimensional Euler equations and related hyperbolic systems of conservation laws is described. Some analytical approaches including techniques, methods and ideas, developed recently, for solving multidimensional steady problems are presented. Some multidimensional unsteady problems are analyzed. Connections between entropy solutions of hyperbolic conservation laws and divergence-measure fields, as well as the theory of divergence-measure fields, are discussed. Some further trends and open problems on the Euler equations and related multidimensional conservation laws are also addressed.

Lagrangian and Eulerian Representations of Fluid Flow

This essay considers the two major ways that the motion of a fluid continuum may be described, either by observing or predicting the trajectories of parcels that are carried about with the flow – which yields a Lagrangian or material representation of the flow — or by observing or predicting the fluid velocity at fixed points in space — which yields an Eulerian or field representation of the flow. Lagrangian methods are often the most efficient way to sample a fluid domain and most of the physical conservation laws begin with a Lagrangian perspective. Nevertheless, almost all of the theory in fluid dynamics is developed in Eulerian or field form. The premise of this essay is that it is helpful to understand both systems, and the transformation between systems is the central theme.

Riemann Invariant Manifolds for the Multidimensional Euler Equations

Siam Journal on Scientific Computing, 1999

A new approach for studying wave propagation phenomena in an inviscid gas is presented. This approach can be viewed as the extension of the method of characteristics to the general case of unsteady multidimensional flow. A family of spacetime manifolds is found on which an equivalent one-dimensional (1-D) problem holds. Their geometry depends on the spatial gradients of the flow, and they provide, locally, a convenient system of coordinate surfaces for spacetime. In the case of zero-entropy gradients, functions analogous to the Riemann invariants of 1-D gas dynamics can be introduced. These generalized Riemann invariants are constant on these manifolds and, thus, the manifolds are dubbed Riemann invariant manifolds (RIM). Explicit expressions for the local differential geometry of these manifolds can be found directly from the equations of motion. They can be space-like or time-like, depending on the flow gradients. This theory is used to develop a second-order unsplit monotonic upstream-centered scheme for conservation laws (MUSCL)-type scheme for the compressible Euler equations. The appropriate RIM are traced back in time, locally, in each cell. This procedure provides the states that are connected with equivalent 1-D problems. Furthermore, by assuming a linear variation of all quantities in each computational cell, it is possible to derive explicit formulas for the states used in the 1-D characteristic problem.