Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in De Lellis and Székelyhidi Jr (Ann Math 170(3):1417–1436, 2009) and Chiodaroli et al. (2013) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes need not converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. We advocate entropy measure-valued solutions, first proposed by DiPerna, as the appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure which constructs stable approximations to entropy measure-valued solutions, and provide sufficient conditions that guarantee that these approximations converge to an entropy measure-valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed paradigm are presented and are utilized to examine several interesting properties of the computed entropy measure-valued solutions.     

Dissipative measure-valued solutions for general conservation laws

In the last years measure-valued solutions started to be considered as a relevant notion of solutions if they satisfy the so-called measure-valued – strong uniqueness principle. This means that they coincide with a strong solution emanating from the same initial data if this strong solution exists. This property has been examined for many systems of mathematical physics, including incompressible and compressible Euler system, compressible Navier-Stokes system et al. and there are also some results concerning general hyperbolic systems. Our goal is to provide a unified framework for general systems, that would cover the most interesting cases of systems, and most importantly, we give examples of equations, for which the aspect of measure-valued – strong uniqueness has not been considered before, like incompressible magnetohydrodynamics and shallow water magnetohydrodynamics.     

Chemotaxis: from kinetic equations to aggregate dynamics

The hydrodynamic limit for a kinetic model of chemotaxis is investigated. The limit equation is a non local conservation law, for which finite time blow-up occurs, giving rise to measure-valued solutions and discontinuous velocities. An adaptation of the notion of duality solutions, introduced for linear equations with discontinuous coefficients, leads to an existence result. Uniqueness is obtained through a precise definition of the nonlinear flux as well as the complete dynamics of aggregates, i.e. combinations of Dirac masses. Finally a particle method is used to build an adapted numerical scheme.     

Shallow water equations: viscous solutions and inviscid limit

We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C ² test-functions, are confined in a compact set in H ^?1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.     

一类拟线性抛物方程组的整体解

本文通过引进熵－熵流时，证明了一类拟线性物方程组Ｃａｕｃｈｙ问题解的整体存在性定理。     

Variational approach for a class of cooperative systems

The aim of this work is to ascertain the characterization of the existence of coexistence states for a class of cooperative systems supported by the study of an associated non-local equation through classical variational methods. Thanks to those results, we are able to obtain the blow-up behavior of the solutions in the whole domain for certain values of the main continuation parameter.     

Measure-valued solution for non-Newtonian compressible isothermal monopolar fluid

The global existence of measure-valued solutions of initial boundary-value problems in bounded domains to systems of partial differential equations for viscous non-Newtonian isothermal compressible monopolar fluid and the global existence of the weak solution for multipolar fluid is proved.     

Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type

Riemann problems with initial data inside elliptic regions are quite different from those for hyperbolic systems. First, we have found that approximate solutions may present persistent oscillations, giving rise to a new type of (measure-valued) waves besides the usual (distributional) ones, shocks and rarefaction waves. Second, any local disturbance of a constant state inside the elliptic region will result in a non-trivial (distributional or, more generally, measure-valued) solution, which is independent of any particular choice of disturbance. For our numerical experiments, we establish two analytical results for testing convergence of finite difference schemes, and for determining expectation values of state functions with respect to the measure-valued solutions when oscillation waves occur. Numerical examples are presented to illustrate those interesting aspects, including the appearance of oscillation waves together with the analysis of the corresponding Young measures.     

Superprocesses with spatial interactions in a random medium

We construct a class of interactive measure-valued diffusions driven by a historical super-Brownian motion and an independent white noise by solving a certain stochastic equation. In doing so, we show that the approach of Perkins (2002) [3] can be used to study the problem examined by Dawson et al. (2001) [1]. This unifies and extends both Dawson et al. (2001) [1] and Perkins (2002) [3] and establishes a new class of measure-valued diffusions. The existence and pathwise uniqueness of the solutions are proved, and the solutions are shown to satisfy the natural martingale problem.     

Multi-dimensional scalar balance laws with discontinuous flux

We consider scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect to both the space variable x and the unknown quantity u. We formulate the definition of entropy weak solutions and provide existence and uniqueness to the considered problem. The problem is formulated in the framework of multi-valued mappings. The notion of entropy measure-valued solutions is used to prove the so-called contraction principle and comparison principle.     

Banach空间中非线性脉冲Volterra型积分方程组的可解性

在较弱的条件下，利用锥理论和单调迭代方法首先建立了Banach空间中一类非线性算子方程组最小最大解的存在性定理；然后作为应用，利用一个新的比较结果，得到了Banach空间中非线性脉冲Volterra型积分方程组的整体解，改进了最近的许多结果．     

非线性非单调二元算子方程组的解及其应用

利用非线性泛函分析中的锥理论和单调迭代的方法,研究了一类非线性非单调二元算子方程组的解的存在性,并给出了收敛于解的迭代序列,然后作为应用,得到了B anach空间中的一类非线性V olterra型积分方程组的解,改进了最近的许多结果.     

动态规划中提出的一类泛函方程组公共解和重合解的存在性定理

本文讨论了动态规划中提出的一类更一般的泛函方程组公共解和重合解的存在性问题.本文的结果不仅包含引文[6,7]中相应结果为特例,而且也对引文[2～5]在讨论动态规划的原理和模型时所提出的一类新型的泛函方程给出解的存在性条件.     

Frequency spanning homoclinic families

A family of maps or flows depending on a parameter ν which varies in an interval, spans a certain property if along the interval this property depends continuously on the parameter and achieves some asymptotic values along it. We consider families of periodically forced Hamiltonian systems for which the appropriately scaled frequency is spanned, namely it covers the semi-infinite line [0,∞). Under some natural assumptions on the family of flows and its adiabatic limit, we construct a convenient labelling scheme for the primary homoclinic orbits which may undergo a countable number of bifurcations along this interval. Using this scheme we prove that a properly defined flux function is C¹ in ν. Combining this proof with previous results of RK and Poje, immediately establishes that the flux function and the size of the chaotic zone depend on the frequency in a non-monotone fashion for a large class of families of Hamiltonian flows.     

Well-posedness of Smoluchowski's coagulation equation for a class of homogeneous kernels

The uniqueness and existence of measure-valued solutions to Smoluchowski's coagulation equation are considered for a class of homogeneous kernels. Denoting by λ∈(-∞,2]?{0} the degree of homogeneity of the coagulation kernel a, measure-valued solutions are shown to be unique under the sole assumption that the moment of order λ of the initial datum is finite. A similar result was already available for the kernels a(x,y)=2, x+y and xy, and is extended here to a much wider class of kernels by a different approach. The uniqueness result presented herein also seems to improve previous results for several explicit kernels. Furthermore, a comparison principle and a contraction property are obtained for the constant kernel.     

一类流动介质下超过程占位时的状态特征

本文考虑流动介质下的粒子系统，在Ｄａｗｓｏｎ假设下引出一类带广义分枝的测度值分枝过程（超过程）．这类超过程描述了更丰富的粒子分枝现象，具有许多独特的性质．在此我们主要研究其积分过程即占位时过程，揭示了它的两个重要特性即密度场的存在性和长时间的收敛性．     

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.     

Infinitely many solutions for elliptic systems with critical exponents

One class of critical growth elliptic systems of two equations is considered on a bounded domain. By using fountain theorem and concentration estimates, we establish the existence of infinitely many solutions in higher values of dimension N?7.     

System of vector quasi-variational inclusions with some applications

In this paper, we consider systems of vector quasi-variational inclusions which include systems of vector quasi-equilibrium problems for multivalued maps, systems of vector optimization problems and several other systems as special cases. We establish existence results for solutions of these systems. As applications of our results, we derive the existence results for solutions of system vector optimization problems, mathematical programs with systems of vector variational inclusion constraints and bilevel problems. Another application of our results provides the common fixed point theorem for a family of lower semicontinuous multivalued maps. Further applications of our results for existence of solutions of systems of vector quasi-variational inclusions are given to prove the existence of solutions of systems of Minty type and Stampacchia type generalized implicit quasi-variational inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.     

Measure-valued solutions and the phenomenon of blow-down in logarithmic diffusion

The existence of solutions of elliptic and parabolic equations with data a measure has always been quite important for the general theory, a prominent example being the fundamental solutions of the linear theory. In nonlinear equations the existence of such solutions may find special obstacles, that can be either essential, or otherwise they may lead to more general concepts of solution. We give a particular review of results in the field of nonlinear diffusion.As a new contribution, we study in detail the case of logarithmic diffusion, associated with Ricci flow in the plane, where we can prove existence of measure-valued solutions. The surprising thing is that these solutions become classical after a finite time. In that general setting, the standard concept of weak solution is not adequate, but we can solve the initial-value problem for the logarithmic diffusion equation in the plane with bounded nonnegative measures as initial data in a suitable class of measure solutions. We prove that the problem is well-posed. The phenomenon of blow-down in finite time is precisely described: initial point masses diffuse into the medium and eventually disappear after a finite time Ti=Mi/4π.