New δ-shock waves in the p-system: a distributional product approach

2019 ◽  
Vol 25 (3) ◽  
pp. 619-629 ◽  
Author(s):  
Adelino Paiva
Keyword(s):  
Weak Solution ◽  
Entire Function ◽  
Shock Waves ◽  
Riemann Problem ◽  
Solution Concept ◽  
P System ◽  
One Dimensional ◽  
System A ◽  
Distributional Product ◽  
Product Approach

This article studies a Riemann problem for the so-called “[Formula: see text]-system”[Formula: see text], [Formula: see text], which rules one-dimensional isentropic thermoelastic media. Such study is made using a product of distributions that allows us to extend both the classical solution concept and a weak solution concept. By considering [Formula: see text] as an entire function that takes real values on the real axis, this product also extends for certain distributions [Formula: see text] the meaning of [Formula: see text]. Under certain conditions, this Riemann problem has solutions that are [Formula: see text]-shock waves. Furthermore, those [Formula: see text]-shock waves satisfy the so-called generalized Rankine–Hugoniot conditions.

GLOBAL WEAK SOLUTIONS OF THE ONE-DIMENSIONAL HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

1993 ◽  
Vol 03 (06) ◽  
pp. 759-788 ◽  
Author(s):  
F. JOCHMANN
Keyword(s):  
Weak Solution ◽  
Viscosity Solutions ◽  
Hydrodynamic Model ◽  
Global Weak Solution ◽  
Limit Problem ◽  
Young Measures ◽  
P System ◽  
Compensated Compactness ◽  
One Dimensional ◽  
The One

The existence of a global weak solution of the one-dimensional hydrodynamic model for semiconductors is proved by the method of artificial viscosity and the theory of compensated compactness. The system is first regularized and global viscosity-solutions are constructed. Letting the viscosity-parameter tend to zero, we obtain a sequence of viscosity-solutions converging in L∞-weak* to a weak solution of the one-dimensional p-system from isoentropic gas dynamics with an electric field term and momentum relaxation. Since the equations are nonlinear and the convergence is only weak, the theory of Young-measures and compensated compactness is applied to obtain a weak solution of the limit problem.

Distributions as initial values in a triangular hyperbolic system of conservation laws

2019 ◽  
Vol 150 (6) ◽  
pp. 2757-2775 ◽  
Author(s):  
C. O. R. Sarrico ◽  
A. Paiva
Keyword(s):  
Cauchy Problem ◽  
Shock Waves ◽  
Conservation Laws ◽  
Classical Solution ◽  
Hyperbolic System ◽  
Initial Conditions ◽  
Solution Concept ◽  
Space Of Distributions ◽  
Consistent Extension ◽  
Distributional Product

AbstractThe present paper concerns the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposing ϕ, ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributions u,v. For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves, δ-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader's convenience, a brief survey of the distributional product is also included.

A DISTRIBUTIONAL PRODUCT APPROACH TO δ-SHOCK WAVE SOLUTIONS FOR A GENERALIZED PRESSURELESS GAS DYNAMICS SYSTEM

2014 ◽  
Vol 25 (01) ◽  
pp. 1450007 ◽  
Author(s):  
C. O. R. SARRICO
Keyword(s):  
Shock Wave ◽  
Gas Dynamics ◽  
General Framework ◽  
Solution Concept ◽  
Singular Solutions ◽  
Wave Solutions ◽  
Shock Wave Solution ◽  
Consistent Extension ◽  
Distributional Product ◽  
Product Approach

In the setting of a product of distributions which is not defined by approximation processes, we are able to consider a Riemann problem for the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0, with the unknown states u, v in convenient spaces of distributions and ϕ, ψ : ℝ → ℝ continuous. A consistent extension of the classical solution concept will show the possible arising of a δ-shock wave solution. This procedure affords a simpler and more general framework to construct singular solutions and can surely be applied to other equations or systems.

scholarly journals Quantitative Experimental and Theoretical Investigations on the Interaction of Shock Waves with Magnetic Fields

1969 ◽  
Vol 24 (10) ◽  
pp. 1449-1457
Author(s):  
H. Klingenberg ◽  
F. Sardei ◽  
W. Zimmermann
Keyword(s):  
Magnetic Fields ◽  
Shock Waves ◽  
Physical Model ◽  
Spectroscopic Method ◽  
Experimental Results ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Theoretical Investigations ◽  
Theoretical Predictions ◽  
Interaction Of Shock Waves

Abstract In continuation of the work on interaction between shock waves and magnetic fields 1,2 the experiments reported here measured the atomic and electron densities in the interaction region by means of an interferometric and a spectroscopic method. The transient atomic density was also calculated using a one-dimensional theory based on the work of Johnson3 , but modified to give an improved physical model. The experimental results were compared with the theoretical predictions.

Propagation of one-dimensional planar and nonplanar shock waves in nonideal radiating gas

2021 ◽  
Vol 33 (4) ◽  
pp. 046106
Author(s):  
Mayank Singh ◽  
Rajan Arora
Keyword(s):  
Shock Waves ◽  
One Dimensional

scholarly journals On a stochastic Burgers equation with Dirichlet boundary conditions

2003 ◽  
Vol 2003 (43) ◽  
pp. 2735-2746 ◽  
Author(s):  
Ekaterina T. Kolkovska
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Martingale Problem ◽  
Weak Limit ◽  
Dirichlet Boundary Conditions ◽  
One Dimensional ◽  
Stochastic Burgers Equation ◽  
The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

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