Game theory of vaccination and depopulation for managing livestock diseases and zoonoses on small-scale farms

Livestock producers adapt their farm management to epidemiological risks in different ways, through veterinary interventions but also by modulating their farm size and the removal rate of animals. The objective of this theoretical study was to elucidate how these behavioral adaptations may affect the epidemiology of highly-pathogenic avian influenza in domestic poultry and the outcome of the implemented control policies. We studied a symmetric population game where the players are broiler poultry farmers at risk of infection and where the between-farms disease transmission is both environmental and mediated by poultry trade. Three types of farmer behaviors were modelled: vaccination, depopulation, and cessation of poultry farming. We found that the transmission level of the disease through trade networks has strong qualitative effects on the system's epidemiological-economic equilibria. In the case of low trade-based transmission, when the monetary cost of infection is high, depopulation behavior can maintain a stable disease-free equilibrium. In addition, vaccination behavior can lead to eradication by private incentives alone – an outcome not seen for human diseases. In a scenario of high trade-based transmission, depopulation behavior has perverse epidemiological effects as it accelerates the spread of disease via poultry trade. In this situation, state interventions should focus on making vaccination technologies available at a low price rather than penalizing infected farms

Existence of Global Weak Solutions for 3D Degenerate Compressible Navier-Stokes Equations

In this paper, we prove the existence of global weak solutions for 3D compressible Navier-Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins entropy conservation. The main contribution of this paper is to derive the Mellet-Vasseur type inequality for the weak solutions, even if it is not verified by the first level of approximation. This provides existence of global solutions in time, for the compressible Navier-Stokes equations, for any $\gamma>1$, in three dimensional space, with large initial data possibly vanishing on the vacuum. This solves an open problem proposed by Lions

Global weak solutions to compressible quantum Navier-Stokes equations with damping

The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations with damping is proved for large data in three dimensional space. The model consists of the compressible Navier-Stokes equations with degenerate viscosity, and a nonlinear third-order differential operator, with the quantum Bohm potential, and the damping terms. The global weak solutions to such system is shown by using the Faedo-Galerkin method and the compactness argument. This system is also a very important approximated system to the compressible Navier-Stokes equations. It will help us to prove the existence of global weak solutions to the compressible Navier-Stokes equations with degenerate viscosity in three dimensional space.Comment: This paper provides the existence of the approximation in arXiv:1501.0680

On the isentropic compressible Navier-Stokes equation

We consider the compressible Navier-Stokes equation with density dependent viscosity coefficients, focusing on the case where those coefficients vanish on vacuum. We prove the stability of weak solutions both in the torus and in the whole space in dimension 2 and 3. The pressure is given by p=rho^gamma, and our result holds for any gamma>1. In particular, we obtain the stability of weak solutions of the Saint-Venant model for shallow water

Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations

We study weak solutions of the 3D Navier-Stokes equations in whole space with $L^2$ initial data. It will be proved that $\nabla^\alpha u$ is locally integrable in space-time for any real $\alpha$ such that $1< \alpha <3$, which says that almost third derivative is locally integrable. Up to now, only second derivative $\nabla^2 u$ has been known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-$L_{loc}^{4/(\alpha+1)}$. These estimates depend only on the $L^2$ norm of initial data and integrating domains. Moreover, they are valid even for $\alpha\geq 3$ as long as $u$ is smooth. The proof uses a good approximation of Navier-Stokes and a blow-up technique, which let us to focusing on a local study. For the local study, we use De Giorgi method with a new pressure decomposition. To handle non-locality of the fractional Laplacian, we will adopt some properties of the Hardy space and Maximal functions.Comment: 62 page

On Uniqueness of Solutions to Conservation Laws Verifying a Single Entropy Condition

For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see Panov [Mat. Zametki, 55(5):116--129, 159, 1994]). This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Quart. Appl. Math., 62(4):687--700, 2004]. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In this paper, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case.Comment: 34 page

L2L^2-contraction for shock waves of scalar viscous conservation laws

We consider the $L^2$-contraction up to a shift for viscous shocks of scalar viscous conservation laws with strictly convex fluxes in one space dimension. In the case of a flux which is a small perturbation of the quadratic burgers flux, we show that any viscous shock induces a contraction in $L^2$, up to a shift. That is, the $L^2$ norm of the difference of any solution of the viscous conservation law, with an appropriate shift of the shock wave, does not increase in time. If, in addition, the difference between the initial value of the solution and the shock wave is also bounded in $L^1$, the $L^2$ norm of the difference converges at the optimal rate $t^{-1/4}$. Both results do not involve any smallness condition on the initial value, nor on the size of the shock. In this context of small perturbations of the quadratic Burgers flux, the result improves the Choi and Vasseur's result in [7]. However, we show that the $L^2$-contraction up to a shift does not hold for every convex flux. We construct a smooth strictly convex flux, for which the $L^2$-contraction does not hold any more even along any Lipschitz shift

De Giorgi Techniques Applied to The Holder Regularity of Solutions to Hamilton-Jacobi Equations

This article is dedicated to the proof of C^{\alpha} regularization effects of Hamilton- Jacobi equations. The proof is based on the De Giorgi method. The regularization is independent on the regularity of the Hamiltonian.Comment: 17 page

Criteria on contractions for entropic discontinuities of systems of conservation laws

We study the contraction properties (up to shift) for admissible Rankine-Hugoniot discontinuities of $n\times n$ systems of conservation laws endowed with a convex entropy. We first generalize the criterion developed in [47], using the spatially inhomogeneous pseudo-distance introduced in [50]. Our generalized criterion guarantees the contraction property for extremal shocks of a large class of systems, including the Euler system. Moreover, we introduce necessary conditions for contraction, specifically targeted for intermediate shocks. As an application, we show that intermediate shocks of the two-dimensional isentropic magnetohydrodynamics do not verify any of our contraction properties. We also investigate the contraction properties, for contact discontinuities of the Euler system, for a certain range of contraction weights. All results do not involve any smallness condition on the initial perturbation, nor on the size of the shock

Stability and uniqueness for piecewise smooth solutions to Burgers-Hilbert among a large class of solutions

In this paper, we show uniqueness and stability for the piecewise-smooth solutions to the Burgers--Hilbert equation constructed in Bressan and Zhang [Commun. Math. Sci., 15(1):165--184, 2017]. The Burgers--Hilbert equation is $u_t+(\frac{u^2}{2})_x=\mathbf{H}[u]$ where $\mathbf{H}$ is the Hilbert transform, a nonlocal operator. We show stability and uniqueness for solutions amongst a larger class than the uniqueness result in Bressan and Zhang. The solutions we consider are measurable and bounded, satisfy at least one entropy condition, and verify a strong trace condition. We do not have smallness assumptions. We use the relative entropy method and theory of shifts (see Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 -- 376, 2008]).Comment: 46 page