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Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations
"... We are concerned with the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as the following secondorde ..."
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Cited by 68 (23 self)
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We are concerned with the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as the following secondorder nonlinear equation of mixed
Smooth Irrotational Flow in the Large to the EulerPoisson System in R 3+1
 in R 3+1 , Comm.Math.Phys
"... : A simple twofluid model to describe the dynamics of a plasma is the EulerPoisson system, where the compressible electron fluid interacts with its own electric field against a constant charged ion background. The plasma frequency produced by the electric field plays the role of `mass' term t ..."
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Cited by 50 (5 self)
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: A simple twofluid model to describe the dynamics of a plasma is the EulerPoisson system, where the compressible electron fluid interacts with its own electric field against a constant charged ion background. The plasma frequency produced by the electric field plays the role of `mass' term to the linearized system. Based on this `KleinGordon' effect, we construct global smooth irrotational flows with small velocity for the electron fluid. 1 Introduction A plasma is a collection of moving electrons and ions. At high frequencies, a simplefluid model for a plasma breaks down. The electrons and ions tend to move independently, and charge separations occur. The greater inertia of the ions implies that they will be unable to follow the rapid fluctuation of the fluid, only electrons partake in the motion. The ions merely provide a uniform background of positive charge. One of the simplest twofluid model for a plasma is the EulerPoisson system @ t n +r \Delta (nu) = 0 (1) @ t u + u...
The Cauchy problem for the Euler equations for compressible
 In: Handbook of Mathematical Fluid Dynamics
, 2002
"... Abstract. Some recent developments in the study of the Cauchy problem for the Euler equations for compressible fluids are reviewed. The local and global wellposedness for smooth solutions is presented, and the formation of singularity is exhibited; then the local and global wellposedness for disco ..."
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Cited by 30 (4 self)
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Abstract. Some recent developments in the study of the Cauchy problem for the Euler equations for compressible fluids are reviewed. The local and global wellposedness for smooth solutions is presented, and the formation of singularity is exhibited; then the local and global wellposedness 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.
Global solutions to the compressible NavierStokes equations for a reacting mixture
 SIAM J. Math. Anal
, 1992
"... We prove the global existence of weak solutions to the NavierStokes equations for compressible, heatconducting flow in one space dimension with large, discontinuous initial data, and we obtain apriori estimates for these solutions which are independent of time, sufficient to determine their asymp ..."
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Cited by 23 (7 self)
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We prove the global existence of weak solutions to the NavierStokes equations for compressible, heatconducting flow in one space dimension with large, discontinuous initial data, and we obtain apriori estimates for these solutions which are independent of time, sufficient to determine their asymptotic behavior. In particular, we show that, as time goes to infinity, the solution tends to a constant state determined by the initial mass and the initial energy, and that the magnitudes of singularities in the solution decay to zero. 1991 Mathematics Subject Classification. 35B40, 35D05, 76N10, 35B45. Key words and phrases. NavierStokes equations, compressible flow, global discontinuous solutions, largetime behavior, large discontinuous initial data, uniform bounds.
The Riemann problem for fluid flows in a nozzle with discontinuous crosssection
 Comm. Math. Sci
"... Abstract. The system of balance laws describing a compressible fluid flow in a nozzle forms a nonstrictly hyperbolic systems of partial differential equations which, also, is not fully conservative due to the effect of the geometry. First, we investigate the general properties of the system and det ..."
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Cited by 15 (8 self)
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Abstract. The system of balance laws describing a compressible fluid flow in a nozzle forms a nonstrictly hyperbolic systems of partial differential equations which, also, is not fully conservative due to the effect of the geometry. First, we investigate the general properties of the system and determine all possible wave combinations. Second, we construct analytically the solutions of the Riemann problem for any values of the left and righthand states. For certain values we obtain up to three solutions whose structure is carefully described here. In some range of Riemann data, no solution exists. When three solutions are available, then exactly one of them contains two stationary waves which are superimposed in the physical space. We include also numerical plots of these solutions. 1.
Global solutions to the NavierStokes equations for compressible heatconducting flow with symmetry and free boundary
 Commun. Partial Diff. Equations
, 2002
"... Global solutions of the multidimensional NavierStokes equations for compressible heatconducting flow are constructed, with spherically symmetric initial data of large oscillation between a static solid core and a free boundary connected to a surrounding vacuum state. The free boundary connects the ..."
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Cited by 13 (0 self)
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Global solutions of the multidimensional NavierStokes equations for compressible heatconducting flow are constructed, with spherically symmetric initial data of large oscillation between a static solid core and a free boundary connected to a surrounding vacuum state. The free boundary connects the compressible heatconducting fluids to the vacuum state with free normal stress and zero normal heat flux. The fluids are initially assumed to fill with a finite volume and zero density at the free boundary, and with bounded positive density and temperature between the solid core and the initial position of the free boundary. One of the main features of this problem is the singularity of solutions near the free boundary. Our approach is to combine an effective difference scheme to construct approximate solutions with the energy methods and the pointwise estimate techniques to deal with the singularity of solutions near the free boundary and to obtain the bounded estimates of the solutions and the free boundary as time evolves. The convergence of the difference scheme is established. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.
A Bound on the Total Variation of the Conserved Quantities for Solutions of General Resonant Nonlinear Balance Laws
 SIAM J. APPLIED MATH
, 2004
"... We introduce a new potential interaction functional and use it to define a new Glimmtype functional that bounds the total variation of the conserved quantities at time t>0by the total variation at time t = 0+ in Glimm approximate solutions of a general resonant nonlinear balance law. ..."
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Cited by 8 (1 self)
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We introduce a new potential interaction functional and use it to define a new Glimmtype functional that bounds the total variation of the conserved quantities at time t>0by the total variation at time t = 0+ in Glimm approximate solutions of a general resonant nonlinear balance law.
Multidimensional conservation laws: Overview, problems, and perspective
 HYPERBOLIC APPROXIMATION OF KINETIC EQUATIONS 549
"... ar ..."
Front tracking for scalar balance equations
 J. Hyperbolic Differ. Equ
"... Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the f ..."
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Cited by 5 (2 self)
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Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL–condition associated with it, and it does not discriminate between stiff and nonstiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady–state solutions (or achieving them in the long time limit) with good accuracy. 1.