-
Previous Article
Models for internal waves in deep water
- DCDS Home
- This Issue
-
Next Article
On the existence of multipeak solutions for nonlinear field equations on $R^N$
BV estimates for multicomponent chromatography with relaxation
| 1. | S.I.S.S.A., Via Beirut, 2-4, 34014 Trieste |
| 2. | S.I.S.S.A., Via Beirut 4, Trieste 34014, Italy |
$u_t + u_x =-\frac{1}{\varepsilon} (F(u)-v),$ (1)
$v_t = \frac{1}{\varepsilon} (F(u)-v),$
This model describes a liquid flowing with unit speed over a solid bed.
Several chemical substances are partly dissolved in the liquid, partly deposited on the solid bed.
Their concentrations are represented respectively by the vectors $u = (u_1, ... , u_n)$ and $v = (v_1, ... , v_n)$.
We show that, if the initial data have small total variation, then the solution of (1)
remains with small variation for all times $t \geq 0$.
Moreover, using the $mathbf L^1$ distance, this
solution depends Lipschitz continuously on the initial data, with a Lipschitz constant
uniform w.r.t. $\varepsilon$.
Finally we prove that as $\varepsilon\to 0$,
the solutions of (1) converge to a
limit described by the system
$(u + F(u))_t + u_x = 0,$ $v= F(u)$. (2)
The proof of the uniform BV estimates relies on the application of probabilistic techniques. It is shown that the components of the gradients $v_x$, $u_x$ can be interpreted as densities of random particles travelling with speed $0$ or $1$. The amount of coupling between different components is estimated in terms of the expected number of crossing of these random particles. This provides a first example where BV estimates are proved for general solutions to a class of $2n \times 2n$ systems with relaxation.
| [1] |
Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4271-4285. doi: 10.3934/dcds.2016.36.4271 |
| [2] |
Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759 |
| [3] |
Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143 |
| [4] |
Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673 |
| [5] |
Gui-Qiang Chen, Monica Torres. On the structure of solutions of nonlinear hyperbolic systems of conservation laws. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1011-1036. doi: 10.3934/cpaa.2011.10.1011 |
| [6] |
Stefano Bianchini. A note on singular limits to hyperbolic systems of conservation laws. Communications on Pure & Applied Analysis, 2003, 2 (1) : 51-64. doi: 10.3934/cpaa.2003.2.51 |
| [7] |
Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717 |
| [8] |
Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185 |
| [9] |
Fumioki Asakura, Andrea Corli. The path decomposition technique for systems of hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 15-32. doi: 10.3934/dcdss.2016.9.15 |
| [10] |
Shijin Deng, Weike Wang. Pointwise estimates of solutions for the multi-dimensional scalar conservation laws with relaxation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1107-1138. doi: 10.3934/dcds.2011.30.1107 |
| [11] |
Mapundi K. Banda, Michael Herty. Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Mathematical Control & Related Fields, 2013, 3 (2) : 121-142. doi: 10.3934/mcrf.2013.3.121 |
| [12] |
Eitan Tadmor. Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4579-4598. doi: 10.3934/dcds.2016.36.4579 |
| [13] |
Stefano Bianchini. On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 329-350. doi: 10.3934/dcds.2000.6.329 |
| [14] |
Tatsien Li, Libin Wang. Global exact shock reconstruction for quasilinear hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 597-609. doi: 10.3934/dcds.2006.15.597 |
| [15] |
Boris Andreianov, Mohamed Karimou Gazibo. Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 203-222. doi: 10.3934/nhm.2016.11.203 |
| [16] |
Yu Zhang, Yanyan Zhang. Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1523-1545. doi: 10.3934/cpaa.2019073 |
| [17] |
Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081 |
| [18] |
Mauro Garavello. A review of conservation laws on networks. Networks & Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565 |
| [19] |
Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159 |
| [20] |
Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]