Formation of Singularities and Existence of Global Continuous Solutions for the Compressible Euler Equations (original) (raw)
Related papers
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 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.
2004
We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singularity formation, ”the best sufficient condition”, in the sense that one can explicitly construct a global in time smooth solution for which this condition is not satisfied ”arbitrary little”. Also compactly supported perturbation of nontrivial constant state is considered. We generalize the known theorem [1] on initial data resulting in singularities. Finally, we investigate the influence of frictional damping and rotation on the singularity formation.
Localization of the formation of singularities in multidimensional compressible Euler equations
arXiv: Analysis of PDEs, 2020
We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial stationary state. We prove the blowup results using the characteristics of the propagation of the solution in space and find upper and lower bounds for the density of a smooth solution in a given region of space in terms of the initial data. To solve the problems, we introduce a special family of integral functionals and study their temporal dynamics.
Journal of Mathematical Sciences, 2007
We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singularity formation, "the best sufficient condition", in the sense that one can explicitly construct a global in time smooth solution for which this condition is not satisfied "arbitrary little". Also compactly supported perturbation of nontrivial constant state is considered. We generalize the known theorem [1] on initial data resulting in singularities. Finally, we investigate the influence of frictional damping and rotation on the singularity formation.
An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations
Communications in Mathematical Physics, 2017
We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier-Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have nonnegative anomalous entropy production and kinetic energy dissipation, with both vanishing when solutions are above the critical degree of Besov regularity. Stationary, planar shocks in Euclidean space with an ideal-gas equation of state provide simple examples that satisfy the conditions of our theorems and which demonstrate sharpness of our L 3-based conditions. These conditions involve space-time Besov regularity, but we show that they are satisfied by Euler solutions that possess similar space regularity uniformly in time.
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.
Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations
2001
We prove the uniqueness of Riemann solutions in the class of entropy solutions in L ∞ ∩ BV loc for the 3 × 3 system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global L 2 -stability of the Riemann solutions even in the class of entropy solutions in L ∞ with arbitrarily large oscillation for the 3 × 3 system. We apply our framework established earlier to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under L 1 perturbation of the Riemann initial data, as long as the corresponding solutions are in L ∞ and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions U (x, t), piecewise Lipschitz in x, for any t > 0.
Global solutions to the compressible Euler equations with geometrical structure
Communications in Mathematical Physics, 1996
We prove the existence of global solutions to the Euler equations of compressible isentropic gas dynamics with geometrical structure, including transonic nozzle flow and spherically symmetric flow. Due to the presence of the geometrical source terms, the existence results themselves are new, especially as they pertain to radial flow in an unbounded region, I~Zl >_-1, and to transonic nozzle flow. Arbitrary data with L ~ bounds are allowed in these results. A shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions. The convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework.
Blow-up phenomena for compressible Euler equations with non-vacuum initial data
Zeitschrift für angewandte Mathematik und Physik, 2015
In this article, we study the blowup phenomena of compressible Euler equations with nonvacuum initial data. Our new results, which cover a general class of testing functions, present new initial value blowup conditions. The corresponding blowup results of the 1-dimensional case in non-radial symmetry are also included.