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Related papers
arXiv: Analysis of PDEs, 2019
We are concerned with the formation of singularities and the existence of global continuous solutions of the Cauchy problem for the one-dimensional non-isentropic Euler equations for compressible fluids. For the isentropic Euler equations, we pinpoint a necessary and sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum --- there exists a compression in the initial data. For the non-isentropic Euler equations, we identify a sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum --- there exists a strong compression in the initial data. Furthermore, we identify two new phenomena --- decompression and de-rarefaction --- for the non-isentropic Euler flows, different from the isentropic flows, via constructing two respective solutions. For the decompression phenomenon, we construct a first global continuous non-isentropic solution, even though initial ...
On regular solutions of the 333D compressible isentropic Euler-Boltzmann equations with vacuum
Discrete and Continuous Dynamical Systems, 2015
In this paper, we discuss the Cauchy problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. We establish the existence of unique local regular solutions with vacuum by the theory of quasi-linear symmetric hyperbolic systems and the Minkowski's inequality under some physical assumptions for the radition quantities. Moreover, it is interesting to show the non-global existence of regular solutions caused by the effect of vacuum for polytropic gases with adiabatic exponent 1 < γ ≤ 3 via some observation on the propagation of the radiation field. Compared with [12][16][22], some new initial conditions that will lead to the finite time blow-up for classical solutions have been introduced. These blow-up results tell us that the radiation effect on the fluid is not strong enough to prevent the formation of singularities caused by the appearance of vacuum.
Global Ill-Posedness of the Isentropic System of Gas Dynamics
Communications on Pure and Applied Mathematics, 2014
We consider the isentropic compressible Euler system in 2 space dimensions with pressure law p(ρ) = ρ 2 and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions.
Compressible Euler Equations¶with General Pressure Law
Archive for Rational Mechanics and Analysis, 2000
We study the hyperbolic system of Euler equations for an isentropic, compressible fluid governed by a general pressure law. The existence and regularity of the entropy kernel that generates the family of weak entropies is established by solving a new Euler-Poisson-Darboux equation, which is highly singular when the density of the fluid vanishes. New properties of cancellation of singularities in combinations of the entropy kernel and the associated entropy-flux kernel are found.
The Cauchy Problem for the Euler Equations for Compressible Fluids
Handbook of Mathematical Fluid Dynamics, 2002
Some recent developments in the study of the Cauchy problem for the Euler equations for compressible fluids are reviewed. The local and global well-posedness for smooth solutions is presented, and the formation of singularity is exhibited; then the local and global well-posedness for discontinuous solutions, including the BV theory and the L ∞ theory, is extensively discussed. Some recent developments in the study of the Euler equations with source terms are also reviewed.
Euler equations for isentropic gas dynamics with general pressure law
Advances in Continuous and Discrete Models
In this work, we explore the limiting behavior of Riemann solutions to the Euler equations in isentropic gas dynamics with general pressure law. We demonstrate that in the distributional sense the delta wave of zero-pressure gas dynamics is formed by a limit solution. Finally, to present the concentration phenomena, we also offer some numerical outcomes.
Advances in Mathematics, 2021
In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations with degenerate viscosities is considered. By introducing some new variables and making use of the " quasi-symmetric hyperbolic"-"degenerate elliptic" coupled structure to control the behavior of the fluid velocity, we prove the global-in-time well-posedness of regular solutions with vacuum for a class of smooth initial data that are of small density but possibly large velocities. Here the initial mass density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial velocity are all positive. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the BD-entropy, and seems to be the first on the global existence of smooth solutions which have large velocities and contain vacuum state for such degenerate system in three space dimensions.
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.
Singularity Formation for the Compressible Euler Equations
SIAM Journal on Mathematical Analysis
It is well-known that singularity will develop in finite time for hyperbolic conservation laws from initial nonlinear compression no matter how small and smooth the data are. Classical results, including Lax [14], John [13], Liu [22], Li-Zhou-Kong [16], confirm that when initial data are small smooth perturbations near constant states, blowup in gradient of solutions occurs in finite time if initial data contain any compression in some truly nonlinear characteristic field, under some structural conditions. A natural question is that: Will this picture keep true for large data problem of physical systems such as compressible Euler equations? One of the key issues is how to find an effective way to obtain sharp enough control on density lower bound, which is known to decay to zero as time goes to infinity for certain class of solutions. In this paper, we offer a simple way to characterize the decay of density lower bound in time, and therefore successfully classify the questions on singularity formation in compressible Euler equations. For isentropic flow, we offer a complete picture on the finite time singularity formation from smooth initial data away from vacuum, which is consistent with the small data theory. For adiabatic flow, we show a striking observation that initial weak compressions do not necessarily develop singularity in finite time. Furthermore, we follow [7] to introduce the critical strength of nonlinear compression, and prove that if the compression is stronger than this critical value, then singularity develops in finite time, and otherwise there are a class of initial data admitting global smooth solutions with maximum strength of compression equals to this critical value.
The Incompressible Limit of the Non-Isentropic Euler Equations
Archive for Rational Mechanics and Analysis, 2001
We study the incompressible limit of classical solutions to the compressible Euler equations for non-isentropic fluids in a domain Ω ⊂ R d. We consider the case of general initial data. For a domain Ω, bounded or unbounded, we first prove the existence of classical solutions for a time independent of the small parameter. Then, in the exterior case, we prove that the solutions converge to the solution of the incompressible Euler equations.