Random conservation laws and global solutions of nonlinear SPDE application to the HJB SPDE of anticipative control

2005 ◽  
pp. 43-53 ◽  
Author(s):  
M. H. A. Davis ◽  
G. Burstein
Keyword(s):  
Conservation Laws ◽  
Global Solutions

Non-classical global solutions for a class of scalar conservation laws

2004 ◽  
Vol 134 (2) ◽  
pp. 225-240
Author(s):  
Paolo Baiti
Keyword(s):  
Conservation Laws ◽  
Global Solutions ◽  
Entropy Inequality ◽  
Travelling Wave ◽  
Scalar Conservation Laws ◽  
Kinetic Relation ◽  
Bounded Variation Functions ◽  
The Cauchy Problem ◽  
Scalar Conservation ◽  
The One

We consider the Cauchy problem for a class of scalar conservation laws with flux having a single inflection point. We prove existence of global weak solutions satisfying a single entropy inequality together with a kinetic relation, in a class of bounded variation functions. The kinetic relation is obtained by the travelling-wave criterion for a regularization consisting of balanced diffusive and dispersive terms. The result is applied to the one-dimensional Buckley-Leverett equation.

scholarly journals An extension of Bakhvalov’s theorem for systems of conservation laws with damping

2017 ◽  
Vol 14 (04) ◽  
pp. 703-719
Author(s):  
Hermano Frid
Keyword(s):  
Periodic Solution ◽  
Initial Data ◽  
Conservation Laws ◽  
Gas Dynamics ◽  
Global Solutions ◽  
Eulerian Formulation ◽  
Systems Of Conservation Laws ◽  
Periodic Initial Data ◽  
Isentropic Gas Dynamics ◽  
Isentropic Gas

For [Formula: see text] systems of conservation laws satisfying Bakhvalov conditions, we present a class of damping terms that still yield the existence of global solutions with periodic initial data of possibly large bounded total variation per period. We also address the question of the decay of the periodic solution. As applications, we consider the systems of isentropic gas dynamics, with pressure obeying a [Formula: see text]-law, for the physical range [Formula: see text], and also for the “non-physical” range [Formula: see text], both in the classical Lagrangian and Eulerian formulation, and in the relativistic setting. We give complete details for the case [Formula: see text], and also analyze the general case when [Formula: see text] is small. Further, our main result also establishes the decay of the periodic solution.

On an Exchange Interaction Model for Quantum Transport: The Schrödinger–Poisson–Slater System

2003 ◽  
Vol 13 (10) ◽  
pp. 1397-1412 ◽  
Author(s):  
Olivier Bokanowski ◽  
José L. López ◽  
Juan Soler
Keyword(s):  
Initial Data ◽  
Conservation Laws ◽  
Exchange Interaction ◽  
Quantum Transport ◽  
Mixed State ◽  
Variational Approach ◽  
Interaction Model ◽  
Global Solutions ◽  
Stationary Solutions ◽  
Exchange Term

We study a mixed-state Schrödinger–Poisson–Slater system (SPSS). This system combines the nonlinear and nonlocal Coulomb interaction with a local potential nonlinearity known as the "Slater exchange term" which models a fermionic effect. The origin of the model is explained and related models are also proposed. Existence, uniqueness and regularity of local-in-time and global solutions are analyzed in ℝ3 with L2 or H1 initial data. Stationary solutions and conservation laws are also analyzed by using a variational approach due to E. Lieb.

scholarly journals Statistical solutions of hyperbolic systems of conservation laws: Numerical approximation

2020 ◽  
Vol 30 (03) ◽  
pp. 539-609 ◽  
Author(s):  
U. S. Fjordholm ◽  
K. Lye ◽  
S. Mishra ◽  
F. Weber
Keyword(s):  
Conservation Laws ◽  
Finite Volume ◽  
Hyperbolic Systems ◽  
Global Solutions ◽  
Monte Carlo Sampling ◽  
Finite Volume Methods ◽  
Sampling Procedure ◽  
Convergence Theory ◽  
Statistical Solution ◽  
Statistical Solutions

Statistical solutions are time-parameterized probability measures on spaces of integrable functions, which have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions are also presented.

Sign in / Sign up

Export Citation Format

Share Document