1,720,969 research outputs found
New interaction estimates for the Baiti-Jenssen system
Full text linkWe 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
An overview on the approximation of boundary Riemann problems through physical viscosity
Full text linkThis note aims at providing an overview of some recent results concerning the viscous approximation of so-called boundary Riemann problems for nonlinear systems of conservation laws in small total variation regimes. © 2016, Sociedade Brasileira de Matemática
Initial–boundary value problems for merely bounded nearly incompressible vector fields in one space dimension
No full textWe establish existence and uniqueness results for initial-boundary value problems for transport equations in one space dimension with nearly incompressible velocity fields, under the sole assumption that the fields are bounded. In the case where the velocity field is either nonnegative or nonpositive, one can rely on similar techniques as in the case of the Cauchy problem. Conversely, in the general case we introduce a new and more technically demanding construction, which heuristically speaking relies on a “lagrangian formulation” of the problem, albeit in a highly irregular setting. We also establish stability of the solution in weak and strong topologies, and propagation of the BV regularity. In the case of either nonnegative or nonpositive velocity fields we also establish a BV-in-time regularity result, and we exhibit a counterexample showing that the result is false in the case of sign-changing vector fields. To conclude, we establish a trace renormalization property
Optimal strategies for a time-dependent harvesting problem
No full textWe focus on an optimal control problem, introduced by Bressan and
Shen in~\cite{BS1} as a model for fish harvesting. We consider the
time-dependent case and we establish existence and uniqueness of an
optimal strategy, and sufficient conditions for optimality. We
also consider a related differential game that models the situation
where there are several competing fish companies and we prove
existence of Nash equilibria. From the technical viewpoint, the
most relevant point is establishing the uniqueness result. This
amounts to prove precise a-priori estimates for solutions of
suitable parabolic equations with measure-valued coefficients. All
the analysis is developed in the case when the fishing domain is
one-dimensional
NEW REGULARITY RESULTS FOR SCALAR CONSERVATION LAWS, AND APPLICATIONS TO A SOURCE-DESTINATION MODEL FOR TRAFFIC FLOWS ON NETWORKS
Full text linkWe focus on entropy admissible solutions of scalar conservation laws in one space dimension and establish new regularity results with respect to time. First, we assume that the flux function f is strictly convex and show that, for every x ∊ R, the total variation of the composite function f ∘ u(⋅, x) is controlled by the total variation of the initial datum. Next, we assume that f is monotone and, under no convexity assumption, we show that, for every x, the total variation of the left and the right trace u(⋅, x±) is controlled by the total variation of the initial datum. We also exhibit a counterexample showing that in the first result the total variation bound does not extend to the function u, or equivalently that in the second result we cannot drop the monotonicity assumption. We then discuss applications to a source-destination model for traffic flows on road networks. We introduce a new approach, based on the analysis of transport equations with irregular coefficients, and, under the assumption that the network only contains so-called T-junctions, we establish existence and uniqueness results for merely bounded data in the class of solutions where the traffic is not congested. Our assumptions on the network and the traffic congestion are basically necessary to obtain well-posedness in view of a counterexample due to Bressan and Yu. We also establish stability and propagation of BV regularity, and this is again interesting in view of recent counterexamples
A connection between viscous profiles and singular ODEs
Full text linkAbstract. We deal with the viscous profiles for a class of mixed hyperbolic-parabolic systems. We focus, in particular, on the case of the compressible Navier Stokes equation in one space variable written in Eulerian coordinates. We describe the link between these profiles and a singular ordinary differential equation in the form dV/dt = F (V )/ζ(V ). Here V ∈ Rd and the function F takes values into Rd and is smooth. The real
valued function ζ is as well regular: the equation is singular in the sense that ζ(V ) can attain the value 0
Some new well-posedness results for continuity and transport equations, and applications to the cromatography system
No full textThe boundary Riemann solver coming from the real vanishing viscosity approximation
No full textWe study the limit of the hyperbolic-parabolic approximation \begin{array}{lll} v_t + \tilde{A} ( v, \, \varepsilon v_x ) v_x = \varepsilon \tilde{B}(v ) v_{xx} \qquad v \in R^N\\ \tilde \beta (v (t, \, 0)) = \bar g \\ v (0, \, x) = \bar v_0. \\ \end{array} \right. The function is defined in such a way to guarantee that the initial boundary value problem is well posed even if is not invertible. The data and are constant. When is invertible, the previous problem takes the simpler form Again, the data and are constant. The conservative case is included in the previous formulations. It is assumed convergence of the v, smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of can be 0. Second, as pointed out before we take into account the possibility that is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur
On the continuum limit of epidemiological models on graphs: Convergence and approximation results
Full text linkWe focus on an epidemiological model (the archetypical SIR system) defined on graphs and study the asymptotic behavior of the solutions as the number of vertices in the graph diverges. By relying on the theory of graphons we provide a characterization of the limit and establish convergence results. We also provide approximation results for both deterministic and random discretizations
Nonlocal Traffic Models with General Kernels: Singular Limit, Entropy Admissibility, and Convergence Rate
Full text linkNonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. While recent counter-examples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement o..
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