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Networks & Heterogeneous Media

2016 , Volume 11 , Issue 2

Special issue on contemporary topics in conservation laws

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Special issue on contemporary topics in conservation laws
Rinaldo M. Colombo , Kenneth H. Karlsen , Frédéric Lagoutière and  Andrea Marson
2016, 11(2): i-ii doi: 10.3934/nhm.2016.11.2i +[Abstract](49) +[PDF](113.0KB)
Abstract:
During last 20 years the theory of Conservation Laws underwent a dramatic development. Networks and Heterogeneous Media is dedicating two consecutive Special Issues to this topic. Researchers belonging to some of the major schools in this subject contribute to these two issues, offering a view on the current state of the art, as well pointing to new research themes within areas already exposed to more traditional methodologies.

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Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws
Boris Andreianov and  Mohamed Karimou Gazibo
2016, 11(2): 203-222 doi: 10.3934/nhm.2016.11.203 +[Abstract](58) +[PDF](642.6KB)
Abstract:
We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definition makes sense under the specific weak trace-regularity assumption. Despite the lack of evidence that generic solutions are trace-regular (especially in space dimension larger than one), the strong entropy formulation may be useful for modeling and numerical purposes.
Relaxation approximation of Friedrichs' systems under convex constraints
Jean-François Babadjian , Clément Mifsud and  Nicolas Seguin
2016, 11(2): 223-237 doi: 10.3934/nhm.2016.11.223 +[Abstract](44) +[PDF](383.8KB)
Abstract:
This paper is devoted to present an approximation of a Cauchy problem for Friedrichs' systems under convex constraints. It is proved the strong convergence in $L^2_{\text{loc}}$ of a parabolic-relaxed approximation towards the unique constrained solution.
Morrey spaces norms and criteria for blowup in chemotaxis models
Piotr Biler , Grzegorz Karch and  Jacek Zienkiewicz
2016, 11(2): 239-250 doi: 10.3934/nhm.2016.11.239 +[Abstract](31) +[PDF](366.7KB)
Abstract:
Two-dimensional Keller--Segel models for the chemotaxis with fractional (anomalous) diffusion are considered. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Similarly, a criterion for blowup of solutions in terms of the radial initial concentrations, related to suitable Morrey spaces norms, is shown for radially symmetric solutions of chemotaxis in several dimensions. Those conditions are, in a sense, complementary to the ones guaranteeing the global-in-time existence of solutions.
Non-critical fractional conservation laws in domains with boundary
Matthieu Brassart
2016, 11(2): 251-262 doi: 10.3934/nhm.2016.11.251 +[Abstract](43) +[PDF](386.3KB)
Abstract:
We study bounded solutions for a multidimensional conservation law coupled with a power $s\in (0,1)$ of the Dirichlet laplacian acting in a domain. If $s \leq 1/2$ then the study centers on the concept of entropy solutions for which existence and uniqueness are proved to hold. If $s >1/2$ then the focus is rather on the $C^\infty$-regularity of weak solutions. This kind of results is known in $\mathbb{R}^N$ but perhaps not so much in domains. The extension given here relies on an abstract spectral approach, which would also allow many other types of nonlocal operators.
New interaction estimates for the Baiti-Jenssen system
Laura Caravenna and  Laura V. Spinolo
2016, 11(2): 263-280 doi: 10.3934/nhm.2016.11.263 +[Abstract](43) +[PDF](504.4KB)
Abstract:
We establish new interaction estimates for a system introduced by Baiti and Jenssen. These estimates are pivotal to the analysis of the wave front-tracking approximation. In a companion paper we use them to construct a counter-example which shows that Schaeffer's Regularity Theorem for scalar conservation laws does not extend to systems. The counter-example we construct shows, furthermore, that a wave-pattern containing infinitely many shocks can be robust with respect to perturbations of the initial data. The proof of the interaction estimates is based on the explicit computation of the wave fan curves and on a perturbation argument.
A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation
Giuseppe Maria Coclite and  Lorenzo di Ruvo
2016, 11(2): 281-300 doi: 10.3934/nhm.2016.11.281 +[Abstract](45) +[PDF](432.8KB)
Abstract:
We consider the Kawahara-Korteweg-de Vries equation, which contains nonlinear dispersive effects. We prove that as the dispersion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the Burgers equation. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.
Logarithmic estimates for continuity equations
Maria Colombo , Gianluca Crippa and  Stefano Spirito
2016, 11(2): 301-311 doi: 10.3934/nhm.2016.11.301 +[Abstract](56) +[PDF](368.4KB)
Abstract:
The aim of this short note is twofold. First, we give a sketch of the proof of a recent result proved by the authors in the paper [7] concerning existence and uniqueness of renormalized solutions of continuity equations with unbounded damping coefficient. Second, we show how the ideas in [7] can be used to provide an alternative proof of the result in [6,9,12], where the usual requirement of boundedness of the divergence of the vector field has been relaxed to various settings of exponentially integrable functions.
A coupling between a non--linear 1D compressible--incompressible limit and the 1D $p$--system in the non smooth case
Rinaldo M. Colombo and  Graziano Guerra
2016, 11(2): 313-330 doi: 10.3934/nhm.2016.11.313 +[Abstract](47) +[PDF](426.6KB)
Abstract:
We consider two compressible immiscible fluids in one space dimension and in the isentropic approximation. The first fluid is surrounded and in contact with the second one. As the sound speed of the first fluid diverges to infinity, we present the proof of rigorous convergence for the fully non--linear compressible to incompressible limit of the coupled dynamics of the two fluids. A linear example is considered in detail, where fully explicit computations are possible.
BV regularity near the interface for nonuniform convex discontinuous flux
Shyam Sundar Ghoshal
2016, 11(2): 331-348 doi: 10.3934/nhm.2016.11.331 +[Abstract](49) +[PDF](448.2KB)
Abstract:
In this paper, we discuss the total variation bound for the solution of scalar conservation laws with discontinuous flux. We prove the smoothing effect of the equation forcing the $BV_{loc}$ solution near the interface for $L^\infty$ initial data without the assumption on the uniform convexity of the fluxes made as in [1,21]. The proof relies on the method of characteristics and the explicit formulas.
On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws
Evgeny Yu. Panov
2016, 11(2): 349-367 doi: 10.3934/nhm.2016.11.349 +[Abstract](36) +[PDF](423.4KB)
Abstract:
We propose a new sufficient non-degeneracy condition for the strong precompactness of bounded sequences satisfying the nonlinear first-order differential constraints. This result is applied to establish the decay property for periodic entropy solutions to multidimensional scalar conservation laws.

2016  Impact Factor: 1.2

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