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This paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions, depending on the sign of the gradient of the solution. We study here the stable case where solutions form a contractive semigroup in L^1. In the spatially periodic case, we prove that semigroup trajectories coincide with the unique limits of a suitable class of vanishing viscosity approximations. 

For a scalar conservation law with strictly convex flux, by Oleinik’s estimates the total variation of a solution with bounded measurable initial data decays like 1/t. This paper introduces a class of intermediate domains where a faster decay rate is achieved. A key ingredient of the analysis is a “Fourier-type” decomposition of u into components which oscillate more and more rapidly. The results aim at extending the theory of fractional domains for analytic semigroups to an entirely nonlinear setting. 

This paper intends to provide a brief review of the current well-posedness theory for hyperbolic systems of conservation laws in one space dimension, also pointing out open problems and possible research directions. 

Given a strictly hyperbolic $$n\times n$$ system of conservation laws, it is well known that there exists a unique Lipschitz semigroup of weak solutions, defined on a domain of functions with small total variation, which are limits of vanishing viscosity approximations. The aim of this note is to prove that every weak solution taking values in the domain of the semigroup, and whose shocks satisfy the Liu admissibility conditions, actually coincides with a semigroup trajectory. 

We consider a class of deterministic mean field games, where the state associated with each player evolves according to an ODE which is linear w.r.t. the control. Existence, uniqueness, and stability of solutions are studied from the point of viewof generic theory. Within a suitable topological space of dynamics and cost functionals, we prove that, for “nearly all” mean field games (in the Baire category sense) the best reply map is single-valued for a.e. player. As a consequence, the mean field game admits a strong (not randomized) solution. Examples are given of open sets of games admitting a single solution, and other open sets admitting multiple solutions. Further examples show the existence of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the best reply map, and another open set of MFG having a unique solution which is unstable. We conclude with an example of a MFG with terminal constraints which does not have any solution, not even in the mild sense with randomized strategies. 

The paper discusses various regularity properties for solutions to a scalar, 1-dimensional conservation law with strictly convex flux and integrable source. In turn, these yield compactness estimates on the solution set. Similar properties are expected to hold for 2x2 genuinely nonlinear systems. 

Abstract Aphids harbor nine common facultative symbionts, most mediating one or more ecological interactions.Wolbachia pipientis, well‐studied in other arthropods, remains poorly characterized in aphids. InPentalonia nigronervosaandP. caladii, global pests of banana,Wolbachiawas initially hypothesized to function as a co‐obligate nutritional symbiont alongside the traditional obligateBuchnera. However, genomic analyses failed to support this role. Our sampling across numerous populations revealed that more than 80% ofPentaloniaaphids carried an M‐supergroup strain ofWolbachia(wPni). The lack of fixation further supports a facultative status forWolbachia, while high infection frequencies in these entirely asexual aphids strongly suggestWolbachiaconfers net fitness benefits. Finding no correlation betweenWolbachiapresence and food plant use, we challengedWolbachia‐infected aphids with common natural enemies. Bioassays revealed thatWolbachiaconferred significant protection against a specialized fungal pathogen (Pandora neoaphidis) but not against generalist pathogens or parasitoids.Wolbachiaalso improved aphid fitness in the absence of enemy challenge. Thus, we identified the first clear benefits for aphid‐associatedWolbachiaand M‐supergroup strains specifically. Aphid‐Wolbachiasystems provide unique opportunities to merge key models of symbiosis to better understand infection dynamics and mechanisms underpinning symbiont‐mediated phenotypes. 

In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi- dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method.