scholarly journals Smooth Approximations of Global in Time Solutions to Scalar Conservation Laws

2009 ◽  
Vol 2009 ◽  
pp. 1-26 ◽  
Author(s):  
V. G. Danilov ◽  
D. Mitrovic
Keyword(s):  
Approximate Solution ◽  
Initial Data ◽  
Conservation Laws ◽  
Asymptotic Method ◽  
Nonlinear Waves ◽  
Conservation Law ◽  
Scalar Conservation Laws ◽  
Scalar Conservation Law ◽  
Scalar Conservation ◽  
Smooth Approximations

We construct global smooth approximate solution to a scalar conservation law with arbitrary smooth monotonic initial data. Different kinds of singularities interactions which arise during the evolution of the initial data are described as well. In order to solve the problem, we use and further develop the weak asymptotic method, recently introduced technique for investigating nonlinear waves interactions.

scholarly journals Stability of the 1D IBVP for a non autonomous scalar conservation law

2018 ◽  
Vol 149 (03) ◽  
pp. 561-592 ◽  
Author(s):  
Rinaldo M. Colombo ◽  
Elena Rossi
Keyword(s):  
Conservation Laws ◽  
Boundary Value Problems ◽  
Conservation Law ◽  
Space Dimension ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Scalar Conservation Law ◽  
Scalar Conservation ◽  
The Stability ◽  
Initial Boundary Value

We prove the stability with respect to the flux of solutions to initial – boundary value problems for scalar non autonomous conservation laws in one space dimension. Key estimates are obtained through a careful construction of the solutions.

scholarly journals EXISTENCE RESULT FOR THE COUPLING PROBLEM OF TWO SCALAR CONSERVATION LAWS WITH RIEMANN INITIAL DATA

2010 ◽  
Vol 20 (10) ◽  
pp. 1859-1898 ◽  
Author(s):  
BENJAMIN BOUTIN ◽  
CHRISTOPHE CHALONS ◽  
PIERRE-ARNAUD RAVIART
Keyword(s):  
Conservation Laws ◽  
Weak Sense ◽  
Scalar Conservation Laws ◽  
Existence Of A Solution ◽  
Coupling Condition ◽  
Coupling Problem ◽  
Flux Function ◽  
Scalar Conservation Law ◽  
Fixed Interface ◽  
Scalar Conservation

This paper is devoted to the coupling problem of two scalar conservation laws through a fixed interface located for instance at x = 0. Each scalar conservation law is associated with its own (smooth) flux function and is posed on a half-space, namely x < 0 or x > 0. At interface x = 0 we impose a coupling condition whose objective is to enforce in a weak sense the continuity of a prescribed variable, which may differ from the conservative unknown (and the flux functions as well). We prove the existence of a solution to the coupled Riemann problem using a constructive approach. The latter allows in particular to highlight interesting features like non-uniqueness of both continuous and discontinuous (at interface x = 0) solutions. The behavior of some numerical scheme is also investigated.

EXISTENCE OF STRONG TRACES FOR GENERALIZED SOLUTIONS OF MULTIDIMENSIONAL SCALAR CONSERVATION LAWS

2005 ◽  
Vol 02 (04) ◽  
pp. 885-908 ◽  
Author(s):  
E. YU. PANOV
Keyword(s):  
Wide Class ◽  
Conservation Law ◽  
Generalized Solutions ◽  
Flux Vector ◽  
Scalar Conservation Laws ◽  
Generalized Entropy ◽  
Scalar Conservation Law ◽  
Nondegeneracy Conditions ◽  
Scalar Conservation ◽  
Class Of Functions

In the half-space t > 0 a multidimensional scalar conservation law with only continuous flux vector is considered. For the wide class of functions including generalized entropy sub- and super-solutions to this equation, we prove existence of the strong trace on the initial hyperspace t = 0. No nondegeneracy conditions on the flux are required.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO CONSERVATION LAWS WITH PERIODIC FORCING

2006 ◽  
Vol 03 (02) ◽  
pp. 387-401 ◽  
Author(s):  
DEBORA AMADORI ◽  
DENIS SERRE
Keyword(s):  
Asymptotic Behavior ◽  
Steady State ◽  
Initial Data ◽  
Conservation Law ◽  
Inflection Point ◽  
Global Minimum ◽  
Asymptotic Behavior Of Solutions ◽  
Periodic Forcing ◽  
Scalar Conservation Law ◽  
Scalar Conservation

We consider the asymptotic behavior of the solution of a forced scalar conservation law, where both the forcing and the initial data are periodic. We prove that there exists a steady state toward which the solution converges in the L1 norm, as t → ∞. We do not assume any smallness or smoothness of the initial data. The limit steady state can be discontinuous, as effect of resonance, and it can be identified when the potential of the forcing term has a unique global minimum, thanks to the conservation of mass. The flux is assumed to be strictly convex; the relevance of this assumption is justified by the construction of solutions with a lattice of periods for a flux with an inflection point.

STRUCTURE OF ENTROPY SOLUTIONS TO SCALAR CONSERVATION LAWS WITH STRICTLY CONVEX FLUX

2012 ◽  
Vol 09 (04) ◽  
pp. 571-611 ◽  
Author(s):  
ADIMURTHI ◽  
SHYAM SUNDAR GHOSHAL ◽  
G. D. VEERAPPA GOWDA
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Conservation Laws ◽  
Compact Support ◽  
Space Dimension ◽  
Structure Theorem ◽  
Regions Of Interest ◽  
Entropy Solutions ◽  
Scalar Conservation Laws ◽  
Scalar Conservation

We consider scalar conservation laws in one space dimension with convex flux and we establish a new structure theorem for entropy solutions by identifying certain shock regions of interest, each of them representing a single shock wave at infinity. Using this theorem, we construct a smooth initial data with compact support for which the solution exhibits infinitely many shock waves asymptotically in time. Our proof relies on Lax–Oleinik explicit formula and the notion of generalized characteristics introduced by Dafermos.

scholarly journals ON THE DIRICHLET-NEUMANN BOUNDARY PROBLEM FOR SCALAR CONSERVATION LAWS

2016 ◽  
Vol 21 (5) ◽  
pp. 685-698
Author(s):  
Marin Mišur ◽  
Darko Mitrovic ◽  
Andrej Novak
Keyword(s):  
Approximate Solution ◽  
Conservation Laws ◽  
Boundary Problem ◽  
Neumann Boundary ◽  
Compensated Compactness ◽  
Scalar Conservation Laws ◽  
Basic Tool ◽  
Numerical Examples ◽  
Elliptic Approximation ◽  
Scalar Conservation

We consider a Dirichlet-Neumann boundary problem in a bounded domain for scalar conservation laws. We construct an approximate solution to the problem via an elliptic approximation for which, under appropriate assumptions, we prove that the corresponding limit satisfies the considered equation in the interior of the domain. The basic tool is the compensated compactness method. We also provide numerical examples.

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