AN ENTROPY BASED GLIMM-TYPE FUNCTIONAL

2008 ◽  
Vol 05 (03) ◽  
pp. 643-662
Author(s):  
LAURA CARAVENNA
Keyword(s):  
Cauchy Problem ◽  
Interaction Potential ◽  
Space Dimension ◽  
Uniformly Convex ◽  
Dimension Formula ◽  
Entropy Dissipation ◽  
Scalar Conservation Law ◽  
The Cauchy Problem ◽  
Type Interaction ◽  
Scalar Conservation

We consider the Cauchy problem for a scalar conservation law in one space dimension [Formula: see text] We introduce, in this simple setting, a new Glimm-type interaction potential: the time marginal of the entropy dissipation measure of a uniformly convex entropy. We show that the Glimm estimates hold for this functional.

scholarly journals Attainable profiles for conservation laws with flux function spatially discontinuous at a single point

2020 ◽  
Vol 26 ◽  
pp. 124
Author(s):  
Fabio Ancona ◽  
Maria Teresa Chiri
Keyword(s):  
Cauchy Problem ◽  
Convex Sets ◽  
Optimization Problems ◽  
Single Point ◽  
Point Of View ◽  
Flux Function ◽  
Full Characterization ◽  
Scalar Conservation Law ◽  
The Cauchy Problem ◽  
Scalar Conservation

Consider a scalar conservation law with discontinuous flux (1): \begin{equation*} \quad u_{t}+f(x,u)_{x}=0, \qquad f(x,u)= \begin{cases} f_l(u)\ &\text{if}\ x<0,\\ f_r(u)\ & \text{if} \ x>0, \end{cases} \quad \quad \quad(1) \end{equation*} where u = u(x, t) is the state variable and fl, fr are strictly convex maps. We study the Cauchy problem for (1) from the point of view of control theory regarding the initial datum as a control. Letting u(x,t)≐StABu-(x) denote the solution of the Cauchy problem for (1), with initial datum u(⋅,0)=u-, that satisfy at x = 0 the interface entropy condition associated to a connection (A, B) (see Adimurthi, S. Mishra and G.D. Veerappa Gowda, J. Hyperbolic Differ. Equ. 2 (2005) 783–837), we analyze the family of profiles that can be attained by (1) at a given time T > 0: \mathcal{A}^{AB}(T)=\left\{\mathcal{S}_T^{AB} \,\overline u : \ \overline u\in{\bf L}^\infty(\mathbb{R})\right\}.\ We provide a full characterization of AAB(T) as a class of functions in BVloc(ℝ\{0}) that satisfy suitable Oleǐnik-type inequalities, and that admit one-sided limits at x = 0 which satisfy specific conditions related to the interface entropy criterion. Relying on this characterisation, we establish the Lloc1-compactness of the set of attainable profiles when the initial data u- vary in a given class of uniformly bounded functions, taking values in closed convex sets. We also discuss some applicationsof these results to optimization problems arising in traffic flow.

scholarly journals On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: Global well-posedness and decay property

2016 ◽  
Vol 13 (03) ◽  
pp. 633-659 ◽  
Author(s):  
Evgeny Yu. Panov
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Almost Periodic Functions ◽  
Entropy Solutions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Scalar Conservation Law ◽  
The Cauchy Problem ◽  
Scalar Conservation ◽  
Necessary And Sufficient

We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux vector in the class of Besicovitch almost periodic functions. The existence and uniqueness of entropy solutions are established. We also uncover the necessary and sufficient condition for the decay of almost periodic entropy solutions as the time variable [Formula: see text]. Our results are then interpreted in the framework of conservation laws on the Bohr compact.

scholarly journals Lagrangian Representations for Linear and Nonlinear Transport

2017 ◽  
Vol 63 (3) ◽  
pp. 418-436
Author(s):  
Stefano Bianchini ◽  
Paolo Bonicatto ◽  
Elio Marconi
Keyword(s):  
Conservation Laws ◽  
Continuity Equation ◽  
Space Dimension ◽  
Nonlinear Transport ◽  
Scalar Conservation Laws ◽  
First Order ◽  
Entropy Dissipation ◽  
Linear And Nonlinear ◽  
Scalar Conservation ◽  
The One

In this note we present a unifying approach for two classes of first order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.

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.

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