Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion

This paper answers to the question whether a shock wave in conservation laws satisfying the Lax shock inequalities but not Oleinik’s entropy criterion is admissible under the vanishing viscosity-capillarity effects. Such a shock appears in van der Waals fluids when a secant line meets the graph of the flux function at four distinct points, and the shock jumps between the two farthest points. The existence of the corresponding traveling waves would justify the admissibility of the shock. For this purpose, we will first show that the corresponding traveling waves satisfy a system of differential equations with two saddle points and two asymptotically stable points. Second, we estimate the domains of attraction of the asymptotically stable equilibrium points, relying on Lyapunov’s stability theory. Third, we investigate the circumstances when an unstable trajectory leaving the saddle point corresponding to the left-hand state of the shock will ever enter the domain of attraction of each of the two asymptotically stable equilibrium points. Finally, we establish the existence of traveling waves associated with a Lax shock but violating the Oleinik’s entropy criterion.     

Existence of traveling wavefronts for Sherratt’s avascular tumor model

In this paper, we revisit Sherratt’s avascular tumor growth system modeled in terms of the continuum densities of proliferating, quiescent and necrotic cells, together with the consideration of the generic nutrient supply from underlying tissue and cell movement of contact inhibition. By adopting a perturbation method combined with the Banach fixed point theorem, we theoretically justify the existence of the traveling wavefronts for this model.     

Existence of solutions to the Riemann problem for 2 × 2 conservation laws

We consider the Riemann problem for a class of 2?×?2 systems of conservation laws which do not satisfy the strictly hyperbolicity condition. Our main assumption is that the product of non-diagonal elements within the F?echet derivative (Jacobian) of the flux is nonnegative. By improving a vanishing viscosity approach, we establish the existence of solutions to the Riemann problem for those systems.     

Existence and uniqueness of solutions to singular ODE’s

We present an existence and uniqueness result for ordinary differential equations based upon the use of weighted spaces. We deal with singular and discontinuous right-hand-sides of the equation.     

Asymptotic Maslov’s method for shocks of conservation laws systems with quadratic flux

In this work we apply the asymptotic method suggested by Maslov [1] to obtain the Hugoniot–Maslov chain for shock type solutions of conservation laws systems with quadratic flux. Additionally to the ODE infinite system that make up the chain, it was obtained an algebraic compatibility condition that must be satisfied by some of the coefficients of the asymptotic expansion of the shock solution. We give a new geometrical interpretation for this compatibility condition by means of certain singular surface whose projections represent time-dependent Hugoniot locus through the left limit state of the Shock.     

Delta shock wave for a 3 × 3 hyperbolic system of conservation laws

We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. The system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that models viscoelastic fluids. An important issue is that the considered 3×3 system is such that every characteristic field is linearly degenerate. We study theRiemann problemfor this system and under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta shock type solutions are established.     

Generalization of Heron’s and Brahmagupta’s equalities to any cyclic polygon

Aequationes mathematicae - It is well known that Heron’s equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been...     

Osgood’s type conditions for uniqueness of entropy solutions to Cauchy problem for quasilinear conservation laws of the first order

In this note, we study the uniqueness problem of generalized solutions in the whole of R ⁿ for conservation systems of the type

An extension of one direction in Marty’s normality criterion

We prove the following extension of one direction in Marty’s theorem: If $k$ is a natural number, $\alpha >1$ and $\mathcal{F }$ is a family of functions meromorphic on a domain $D$ all of whose poles have multiplicity at least $\frac{k}{\alpha -1}$ , then the normality of $\mathcal{F }$ implies that the family $$\begin{aligned} \left\{ \frac{|f^{(k)}|}{1+|f|^\alpha }\,:\, f\in \mathcal{F }\right\} \end{aligned}$$ is locally uniformly bounded.     

Minimal-speed selection of traveling waves to the Lotka–Volterra competition model

In this paper the minimal-speed determinacy of traveling wave fronts of a two-species competition model of diffusive Lotka–Volterra type is investigated. First, a cooperative system is obtained from the classical Lotka–Volterra competition model. Then, we apply the upper-lower solution technique on the cooperative system to study the traveling waves as well as its minimal-speed selection mechanisms: linear or nonlinear. New types of upper and lower solutions are established. Previous results for the linear speed selection are extended, and novel results on both linear and nonlinear selections are derived.     

Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate method for a general system of conservation laws. Furthermore, we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws. Convergence to entropy solutions for scalar conservation laws and for linear symmetrizable systems is also shown. Numerical experiments are presented to illustrate the robustness of the proposed schemes.     

The reason of Hopf’s and Oleinik’s proofs for countability of shocks being wrong

For the number of complete shock curves of a conservation law with one space variable,Hopf in 1950 for the Burger equation,and Oleinik in 1956 for the general,stated that it is at most countable.In 1979,the present author published an example to show that the statement of Hopf and Oleinik is wrong.But after so long time,the wrong statement for countability still appeared in some publications,which is at least partly due to that some ones felt difficult to understand Hopf and Oleinik’s proofs being wrong.So,pointing out where they went wrong becomes very necessary.     

A note on poroacoustic traveling waves under Darcy’s law: Exact solutions

A mathematical analysis of poroacoustic traveling wave phenomena is presented. Assuming that the fluid phase satisfies the perfect gas law and that the drag offered by the porous matrix is described by Darcy’s law, exact traveling wave solutions (TWS)s, as well as asymptotic/approximate expressions, are derived and examined. In particular, stability issues are addressed, shock and acceleration waves are shown to arise, and special/limiting cases are noted. Lastly, connections to other fields are pointed out and possible extensions of this work are briefly discussed.     

Existence and conservation laws for the Boltzmann–Fermi–Dirac equation in a general domain

We prove an existence theorem for the Boltzmann–Fermi–Dirac equation for integrable collision kernels in possibly bounded domains with specular reflection at the boundaries, using the characteristic lines of the free transport. We then obtain that the solution satisfies the local conservations of mass, momentum and kinetic energy thanks to a dispersion technique.     

An extension of Babbage’s criterion for primality

Let n > 1 and k > 1 be positive integers. We show that if $$\left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right) \equiv 1 (\bmod k)$$ for each integer m with 0 ≤ m ≤ n ? 1, then k is a prime and n is a power of this prime. In particular, this assertion under the hypothesis that n = k implies that n is a prime. This was proved by Babbage, and thus our result may be considered as a generalization of this criterion for primality.     

A refinement of Nesterenko’s linear independence criterion with applications to zeta values

We refine (and give a new proof of) Nesterenko’s famous linear independence criterion from 1985, by making use of the fact that some coefficients of linear forms may have large common divisors. This is a typical situation appearing in the context of hypergeometric constructions of mathbbQ{mathbb{Q}}-linear forms involving zeta values or their q-analogs. We apply our criterion to sharpen previously known results in this direction.     

On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation

In this paper we study the existence and qualitative properties of traveling waves associated with a nonlinear flux limited partial differential equation coupled to a Fisher–Kolmogorov–Petrovskii–Piskunov type reaction term. We prove the existence and uniqueness of finite speed moving fronts of C²$C^2$ classical regularity, but also the existence of discontinuous entropy traveling wave solutions.