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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 to the ..."
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Cited by 25 (2 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 14 (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.
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 8 (3 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.
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 5 (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.
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 3 (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.
Riemann Problems With A Kink
"... We study the Riemann problem for isothermal flow of a gas in a thin pipe with a kink in it. This is modeled by a 2 \Theta 2 system of conservation laws with Dirac measure sink term concentrated at the location of the bends in the pipe. We show that the Riemann problem for this system of equations al ..."
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Cited by 2 (0 self)
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We study the Riemann problem for isothermal flow of a gas in a thin pipe with a kink in it. This is modeled by a 2 \Theta 2 system of conservation laws with Dirac measure sink term concentrated at the location of the bends in the pipe. We show that the Riemann problem for this system of equations always has a unique solution, given an extra condition relating the speeds on both sides of the kink. Furthermore, we study the related problem where the flow is perturbed by an continuous addition of momentum at distinct points. Under certain conditions we show that also this Riemann problem has a unique solution. August 19, 1998 0.
Hyperbolic Conservation Laws with a Moving Source
, 1997
"... The purpose of this paper is to investigate the wave behavior of hyperbolic conservation laws with a moving source. When the speed of the source is close to one of the characteristic speeds of the system, nonlinear resonance occurs and instability may result. We will study solutions with a single tr ..."
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The purpose of this paper is to investigate the wave behavior of hyperbolic conservation laws with a moving source. When the speed of the source is close to one of the characteristic speeds of the system, nonlinear resonance occurs and instability may result. We will study solutions with a single transonic shock wave for a general system u t + f(u) x = g(x; u). Suppose that the ith characteristic speed is close to zero. We propose the following stability criterion: l i @g @u r i ! 0 for nonlinear stability, l i @g @u r i ? 0 for nonlinear instability Here l i and r i are the ith normalized left and right eigenvectors of df du respectively. By using a variation of the Glimm scheme and studying the evolution of the single transonic shock wave, we prove the existence of solutions and verify the asymptotic stability (or instability). 1 Introduction In this paper, we study the timeasymptotic stability and instability of solutions to systems of conservation laws with a moving sourc...
Symmetric Euler and NavierStokes shocks in stationary barotropic flow on a bounded domain
, 2008
"... We construct stationary solutions to the barotropic, compressible Euler and NavierStokes equations in several space dimensions with spherical or cylindrical symmetry. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations ..."
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Cited by 1 (1 self)
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We construct stationary solutions to the barotropic, compressible Euler and NavierStokes equations in several space dimensions with spherical or cylindrical symmetry. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations in the exterior domain. On the other hand, stationary smooth solutions in the interior domain necessarily become sonic and can not be continued beyond a critical inner radius. We then use these solutions to construct entropysatisfying shocks for the Euler equations in the region between two concentric spheres or cylinders. Next we construct smooth NavierStokes solutions converging to the previously constructed Euler shocks in the small viscosity limit. In the process we introduce a new technique for constructing smooth solutions, which exhibit a fast
THE GENERIC SOLUTION OF THE RIEMANN PROBLEM IN A NEIGHBORHOOD OF A POINT OF RESONANCE FOR SYSTEMS OF NONLINEAR BALANCE LAWS
, 2003
"... We describe the generic solution of the Riemann problem near a point of resonance in a general 2x2 system of balance laws coupled to a stationary source. The source is treated as a conserved quantity in an augmented 3x3 system, and Resonance is between a nonlinear wave family and the stationary sou ..."
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Cited by 1 (0 self)
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We describe the generic solution of the Riemann problem near a point of resonance in a general 2x2 system of balance laws coupled to a stationary source. The source is treated as a conserved quantity in an augmented 3x3 system, and Resonance is between a nonlinear wave family and the stationary source. Transonic compressible Euler flow in a variable area duct, as well as spherically symmetric flow, are shown to be special cases of the general class of equations studied here.