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.

Semidirect Product Compactifications

1983 ◽  
Vol 35 (1) ◽  
pp. 1-32
Author(s):  
F. Dangello ◽  
R. Lindahl
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Topological Semigroup ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Topological Semigroups ◽  
Weakly Almost Periodic ◽  
Necessary And Sufficient

1. Introduction. K. Deleeuw and I. Glicksberg [4] proved that if S and T are commutative topological semigroups with identity, then the Bochner almost periodic compactification of S × T is the direct product of the Bochner almost periodic compactifications of S and T. In Section 3 we consider the semidirect product of two semi topological semigroups with identity and two unital C*-subalgebras and of W(S) and W(T) respectively, where W(S) is the weakly almost periodic functions on S. We obtain necessary and sufficient conditions and for a semidirect product compactification of to exist such that this compactification is a semi topological semigroup and such that this compactification is a topological semigroup. Moreover, we obtain the largest such compactifications.

Transport-collapse scheme for heterogeneous scalar conservation laws

2018 ◽  
Vol 15 (01) ◽  
pp. 119-132
Author(s):  
Darko Mitrović ◽  
Andrej Novak
Keyword(s):  
Cauchy Problem ◽  
Kinetic Equation ◽  
Conservation Laws ◽  
Admissible Solution ◽  
Scalar Conservation Laws ◽  
Numerical Examples ◽  
The Cauchy Problem ◽  
Scalar Conservation

We extend Brenier’s transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws i.e. for the conservation laws with spacetime-dependent coefficients. The method is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. We also provide numerical examples.

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.

THE CONNECTION BETWEEN PSEUDO ALMOST PERIODIC FUNCTIONS DEFINED ON TIME SCALES AND ON THE REAL LINE

2017 ◽  
Vol 95 (3) ◽  
pp. 482-494 ◽  
Author(s):  
CHAO-HONG TANG ◽  
HONG-XU LI
Keyword(s):  
Time Scales ◽  
Almost Periodic Functions ◽  
Almost Periodic Function ◽  
Almost Periodic ◽  
Necessary And Sufficient Condition ◽  
Periodic Functions ◽  
Pseudo Almost Periodic Functions ◽  
Necessary And Sufficient ◽  
Pseudo Almost Periodic ◽  
Uniformly Continuous

A necessary and sufficient condition for a continuous function $g$ to be almost periodic on time scales is the existence of an almost periodic function $f$ on $\mathbb{R}$ such that $f$ is an extension of $g$. Our aim is to study this question for pseudo almost periodic functions. We prove the necessity of the condition for pseudo almost periodic functions. An example is given to show that the sufficiency of the condition does not hold for pseudo almost periodic functions. Nevertheless, the sufficiency is valid for uniformly continuous pseudo almost periodic functions. As applications, we give some results on the connection between the pseudo almost periodic (or almost periodic) solutions of dynamic equations on time scales and of the corresponding differential equations.

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