Linear stability of shock profiles for a quasilinear Benney system in ℝ2 × ℝ+

2020 ◽  
Vol 17 (04) ◽  
pp. 797-807
Author(s):  
João-Paulo Dias
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Short Wave ◽  
Wave Interactions ◽  
Hyperbolic Conservation ◽  
Linearized Stability ◽  
Shock Profiles ◽  
Scalar Conservation ◽  
Semilinear Schrödinger Equation ◽  
Shock Profile

Following Dias et al. [Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws, J. Differential Equations 251 (2007) 555–563], we study the linearized stability of a pair [Formula: see text], where [Formula: see text] is a shock profile for a family of quasilinear hyperbolic conservation laws in [Formula: see text] coupled with a semilinear Schrödinger equation.

VISCOUS SHOCK PROFILES FOR 2 × 2 SYSTEMS OF HYPERBOLIC CONSERVATION LAWS WITH AN UMBILIC POINT

2009 ◽  
Vol 06 (03) ◽  
pp. 483-524
Author(s):  
FUMIOKI ASAKURA ◽  
MITSURU YAMAZAKI
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Propagation Speed ◽  
Umbilic Point ◽  
Hyperbolic Conservation ◽  
Reservoir Flow ◽  
Viscous Profile ◽  
Systems Of Conservation Laws ◽  
Shock Profiles ◽  
Shock Profile

This article analyzes the existence of viscous shock profiles joining two states satisfying the Rankine–Hugoniot condition that comes from hyperbolic 2 × 2 systems of conservation laws having quadratic flux functions with an isolated umbilic point: the point where the characteristic speeds coincide and the Jacobian matrix of the flux functions is diagonalizable. The systems studied in this note are particularly in Schaeffer and Shearer's cases I and II which are relevant to the three-phase Buckley–Leverett model for oil reservoir flow. It is shown that any compressive and overcompressive shocks have a viscous shock profile provided that there are no undercompressive shock with viscous profile having the same propagation speed. The idea of the proof is a generalization of the first theorem of Morse to noncompact level sets. It is also shown that there exists a shock satisfying the Liu–Oleĭnik condition but having no viscous shock profile. In this case, there is an undercompressive shock with viscous shock profile.

scholarly journals Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds

2019 ◽  
Vol 16 (03) ◽  
pp. 519-593
Author(s):  
L. Galimberti ◽  
K. H. Karlsen
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Generalized Solutions ◽  
Well Posedness ◽  
Scalar Conservation Laws ◽  
Stochastic Forcing ◽  
Rigidity Result ◽  
Hyperbolic Conservation ◽  
The Cauchy Problem ◽  
Scalar Conservation

We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Itô) noise. The Cauchy problem defined on a Riemanian manifold is shown to be well-posed. We prove existence of generalized kinetic solutions using the vanishing viscosity method. A rigidity result àla Perthame is derived, which implies that generalized solutions are kinetic solutions and that kinetic solutions are uniquely determined by their initial data ([Formula: see text] contraction principle). Deprived of noise, the equations we consider coincide with those analyzed by Ben-Artzi and LeFloch [Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 24(6) (2007) 989–1008], who worked with Kružkov–DiPerna solutions. In the Euclidian case, the stochastic equations agree with those examined by Debussche and Vovelle [Scalar conservation laws with stochastic forcing, J. Funct. Anal. 259(4) (2010) 1014–1042].

LARGE TIME BEHAVIOR OF NUMERICAL SOLUTIONS OF SCALAR CONSERVATION LAWS

2006 ◽  
Vol 03 (04) ◽  
pp. 631-648
Author(s):  
FRÉDÉRIC LAGOUTIÈRE
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Large Time ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Hyperbolic Conservation ◽  
Periodic Initial Data ◽  
Scalar Conservation

We study the large time behavior of entropic approximate solutions to one-dimensional, hyperbolic conservation laws with periodic initial data. Under mild assumptions on the numerical scheme, we prove the asymptotic convergence of the discrete solutions to a time- and space-periodic solution.

scholarly journals Optimal jump set in hyperbolic conservation laws

2020 ◽  
Vol 17 (04) ◽  
pp. 765-784
Author(s):  
Shyam Sundar Ghoshal ◽  
Animesh Jana
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Entropy Solution ◽  
Hyperbolic Systems ◽  
Scalar Case ◽  
Qualitative Properties ◽  
Hyperbolic Conservation ◽  
Scalar Conservation Law ◽  
Scalar Conservation ◽  
Jump Set

We investigate qualitative properties of entropy solutions to hyperbolic conservation laws, and construct an entropy solution to a scalar conservation law for which the jump set is not closed, in particular, it is dense in a space-time domain. In a second part, we establish a similar result for hyperbolic systems. We give two different approaches for scalar conservation laws and hyperbolic systems in order to obtain these results. For the scalar case, the solutions are explicitly calculated.

scholarly journals High order explicit local time stepping methods for hyperbolic conservation laws

2020 ◽  
Vol 89 (324) ◽  
pp. 1807-1842
Author(s):  
Thi-Thao-Phuong Hoang ◽  
Lili Ju ◽  
Wei Leng ◽  
Zhu Wang
Keyword(s):  
Conservation Laws ◽  
Local Time ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Hyperbolic Conservation ◽  
Time Stepping ◽  
Local Time Stepping
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