Initial boundary value problems for hyperbolic systems

Kreiss, Heinz–Otto

doi:10.1002/cpa.3160230304

Cited by 701 publications

(692 citation statements)

References 4 publications

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“…[15]) do not apply. Indeed, characteristic speeds of the compressible isentropic Euler equations become singular with infinite spatial derivatives at vacuum boundaries which creates much severe difficulties in analyzing the regularity near boundaries.…”

Section: γ(T) = U(γ(t) T)mentioning

confidence: 99%

Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping

Luo

Zeng

2015

Comm Pure Appl Math

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For the physical vacuum free boundary problem with the sound speed being C 1/2 -Hölder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the the porous media equation with the same total mass when the initial data is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in supreme norm and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher-order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher-order nonlinear energy estimates and elliptic estimates.

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Section: γ(T) = U(γ(t) T)mentioning

confidence: 99%

Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping

Luo

Zeng

2015

Comm Pure Appl Math

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“…For hyperbolic equations it is well-known that the number of boundary conditions at each boundary should be equal to the number of the characteristics directed into the domain at this point [10, p. 417] and [7,15]. An easy way to understand this follows from the concept of "domain of dependence" [3, p. 438-449].…”

Section: Boundary Conditions For the Jet Shapementioning

confidence: 99%

On Modeling of Curved Jets of Viscous Fluid Hitting a Moving Surface

Hlod

2012

Mathematics in Industry

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“…Once the characteristic equation (1.1), which is a polynomial equation with respect to 5, has been determined, the difficulty is reduced to the location of the solutions of this equation in the complex plane. The case of weakly well-posed problems corresponds to the result of Hersch [5], while the case of strongly well-posed problems corresponds to the result of Kreiss [10]. These results can be summarized as follows Weak well-posedness •& {solutions of (1.1)} c {Res > 0}, Strong well-posedness •» {solutions of (1.1)} c {Res > 0}.…”

Section: Introductionmentioning

confidence: 95%

“…The theory of the stability of initial boundary value problems for hyperbolic systems underwent an important development at the beginning of the 1970s with the major work of Kreiss [10]. A very interesting review paper of this theory has been recently given by Higdon in [8].…”

Section: Introductionmentioning

confidence: 99%

“…The other major interest of the concept of strong well-posedness is the fact that Kreiss was unable to prove that the stability results could be extended to smooth variable coefficients and lower-order perturbations. His proof is very complicated and makes use of the theory of symmetrizers and pseudodifferential operators ( [10]). The precise condition about the variations of the coefficients (which are allowed to vary in space and time) can be stated as follows:…”

Section: Introductionmentioning

confidence: 99%

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On the stability analysis of boundary conditions for the wave equation by energy methods. I. The homogeneous case

Ha-Duong¹,

Joly²

1994

Math. Comp.

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Abstract. We reconsider the stability theory of boundary conditions for the wave equation from the point of view of energy techniques. We study, for the case of the homogeneous half-space, a large class of boundary conditions including the so-called absorbing conditions. We show that the results of strong stability in the sense of Kreiss, studied from the point of view of the modal analysis by Trefethen and Halpern, always correspond to the decay in time of a particular energy. This result leads to the derivation of new estimates for the solution of the associated mixed problem.

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Initial boundary value problems for hyperbolic systems

Cited by 701 publications

References 4 publications

Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping

Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping

On Modeling of Curved Jets of Viscous Fluid Hitting a Moving Surface

On the stability analysis of boundary conditions for the wave equation by energy methods. I. The homogeneous case

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