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Saturday, August 8, 2020 | History

4 edition of **Nonlinear stability of shock waves for viscous conservation laws** found in the catalog.

- 146 Want to read
- 11 Currently reading

Published
**1985**
by American Mathematical Society in Providence, R.I., USA
.

Written in

- Viscous flow.,
- Conservation laws (Physics),
- Shock waves.,
- Stability.

**Edition Notes**

Bibliography.

Statement | Tai-Ping Liu. |

Series | Memoirs of the American Mathematical Society -- no.328 |

Classifications | |
---|---|

LC Classifications | QA3, QA929 |

The Physical Object | |

Pagination | p. cm |

ID Numbers | |

Open Library | OL22638782M |

ISBN 10 | 0821823299 |

L2-CONTRACTION OF LARGE PLANAR SHOCK WAVES FOR MULTI-DIMENSIONAL SCALAR VISCOUS CONSERVATION LAWS MOON-JIN KANG, ALEXIS F. VASSEUR, AND YI WANG Abstract. We consider a L2-contraction (a L2-type stability) of large viscous shock waves for the multi-dimensional scalar viscous conservation laws, up to a suitable shift. detonation waves, which have been investigated in [9, 28], and interfaces that connect wave trains in reaction-di usion systems, which have been analyzed, for instance, in [23]. The purpose of this paper is to prove that spectral stability of time-periodic waves in viscous conservation laws implies their nonlinear stability. In the.

This paper is concerned with nonlinear stability of viscous shock profiles for the one-dimensional isentropic compressible Navier-Stokes equations. For the case when the diffusion wave introduced in [6, 7] is excluded, such a problem has been studied in [5, 11] and local stability of weak viscous shock profiles is well-established, but for the corresponding result with large initial. The stability theory of "viscous" shock waves has received a new, geometric perspective due to the work of Kevin Zumbrun and collaborators, which offers a spectral approach to systems. Due to the intersection of point and essential spectrum, such an ap proach had for a long time seemed out of reach.

viscous shock wave nonlinear stability large time asymptotic stability wave speed oe doe grant de-fger nonlinear viscous conservation law gamma oet sigma1 oe gamma gamma constant state sigma viscous p-shock wave solution wave solution royal institute rankine-hugoniot condition. Pointwise asymptotic behavior of perturbed viscous shock profiles Howard, Peter and Raoofi, Mohammadreza, Advances in Differential Equations, ; Survey of admissible shock waves for $2 \times 2$ systems of conservation laws with an umbilic point Asakura, Fumioki and Yamazaki, Mitsuru,, ; On the stability of viscous shock fronts for certain conservation laws in two-dimensional .

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Title (HTML): Nonlinear Stability of Shock Waves for Viscous Conservation Laws Author(s) (Product display): Tai-Ping Liu Book Series Name: Memoirs of. On the stability of viscous shock fronts for certain conservation laws in two-dimensional space Nishikawa, Masataka, Differential and Integral Equations, Viscous limits for a Riemannian problem to a class of systems of conservation laws Zhang, Yanyan and Zhang, Yu, Rocky Mountain Journal of Mathematics, Cited by: In this paper we establish the nonlinear stability of shock waves for viscous conservation laws.

It is shown that when the initial data is a perturbation of viscous shock waves, then the solution converges to viscous shock waves, properly translated, as time tends to infinity. Contributions of more theoretical nature cover the following topics: global existence and uniqueness theory of one-dimensional systems, multidimensional conservation laws in several space variables and approximations of their solutions, mathematical analysis of fluid motion, stability and dynamics of viscous shock waves, singular limits for.

Heinrich Freistühler and Tai-Ping Liu, Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws, Comm. Math. Phys. (), no. 1, – MR (94f). Our stability analysis is strongly motivated by the nonlinear stability of the viscous shock pro le for viscous conservation laws of the form ().

There have been extensive studies in the last three decades [7, 11, 17, 26]. Recently, some important papers on developing the stability theory for nonconvex equations (see [12, 16, 10, 24]) appeared. Our stability analysis is strongly motivated by the nonlinear stability of viscous shock waves for systems of conservation laws with viscosity of the form ut + f(U)x = VUxx, v > 0.

() There have been extensive studies of this question in the last three decades [5, 6, 11, 15, 22, 29]. Cite this paper as: Zumbrun K. () Stability and Dynamics of Viscous Shock Waves. In: Bressan A., Chen GQ., Lewicka M., Wang D. (eds) Nonlinear Conservation Laws. This is an expository article containing a brief overview of key issues related to the stability of nonlinear waves, an introduction to a particular technique in stability analysis known as pointwise estimates, and two applications of this technique: time-periodic shocks in viscous conservation laws [ 3] and source defects in reaction diffusion equations [ 1, 2 ].

Stability and dynamics of viscous shock waves Kevin Zumbrun∗ J IMA Summer School Lectures in Conservation Laws: preliminary version Abstract We examine from a classical dynamical systems point of view stability, dynamics, and bifurcation of viscous shock waves and related solutions of nonlinear pde.

The. The large time asymptotic stability of solutions for systems of nonlinear viscous conservation laws are studied. Systems which are strictly hyperbolic and either genuinely nonlinear or linear degenerate in each characteristic field are addressed.

Such systems possess a smooth travelling wave solution, which is called a viscous p-shock wave solution, provided that the shock strength epsilon is.

Heinrich Freistühler, Tai-Ping Liu, Nonlinear stability of overcompresive shock waves in a rotationally invariant system of viscous conservation laws, Communications in Mathematical Physics, /BF,1, (), ().

Genre/Form: Electronic books: Additional Physical Format: Print version: Liu, Tai-Ping, Nonlinear stability of shock waves for viscous conservation laws /. The central issue throughout the book is the understanding of nonlinear wave interactions.

The book describes the qualitative theory of shock waves. It begins with the basics of the theory for scalar conservation law and Lax's solution of the Reimann problem.

In the present paper, it is shown that the large amplitude viscous shock wave is nonlinearly stable for isentropic Navier-Stokes equations, in which the pressure could be general and includes γ-law, and the viscosity coefficient is a smooth function of density.

The strength of shock wave could be arbitrarily large. of nonlinear waves and their Green’s function using the Hopf-Cole transfor-mation, [8], [2]. We then consider the general convex viscous conservation law ut +f(u)x = κuxx, f′′(u) 6= 0, u∈ R.

For these conservation laws, there is the energy method for the study of the stability of nonlinear waves. The present authors have initiated. gument for nonlinear stability of viscous shock waves developed for general systems of conservation laws in [Z1, MaZ2, MaZ3, MaZ4], based on instantaneous tracking of the lo- cation of the perturbed viscous shock wave.

The advantage of Burgers equation is that. We study the nonlinear stability of shock waves for viscous conservation laws. Our approach is based on a new construction of a fundamental solution for a linearized system around a shock profile.

We consider the L 2-contraction up to a shift for viscous shocks of scalar viscous conservation laws with strictly convex fluxes in one space the case of a flux which is a small perturbation of the quadratic Burgers flux, we show that any viscous shock induces a contraction in L 2, up to a is, the L 2 norm of the difference of any solution of the viscous conservation.

Sufficient conditions for nonlinear stability of viscous shock wave solutions of systems of conservation laws are given. The analysis applies to strong shocks of Lax type but is restricted to perturbations with zero mass.

We use the Laplace transform and reduce the question of stability to a spectral condition for the resolvent equation of the linearized problem. Energy Estimates for Nonlinear Conservation Laws with the energy method for stability analysis. It is shown that if the scalar non-linear shock operators to account for the dissipation of energy by the shock waves.

In the case of the viscous Burgers equation, it is also shown that shock waves .Extending results of Oh and Zumbrun and of Johnson and Zumbrun for parabolic conservation laws, we show that spectral stability implies nonlinear stability for spatially periodic viscous roll wave solutions of the one-dimensional St.

Venant equations for shallow water flow down an inclined ramp.] VISCOUS RAREFACTION WAVES location needs to be identiﬁed exactly for the perturbation to decay around it. Thus the stability of a shock wave is necessarily locally in L1(x).

The shock location can either be a priori determined through global conservation laws or be traced using local conservation laws, see [9], [10], [11], [13], and.