Young measure solutions of some nonlinear mixed‐type equations

This contribution deals with measure‐valued solutions to two types of nonlinear partial differential equations for which, in general, the results on the existence of classical or weak solutions fail. These are the potential equation for transonic flow and the associated unsteady problem (forward–backward diffusion equation). The solutions are constructed by an iteration scheme (Katchanov method) and additional time discretization (Rothe method) in the second case. The existence is proved in the sense of spatial gradient Young measures. Copyright © 2010 John Wiley & Sons, Ltd.     

Orlicz–Young Structures of Young Measures

The paper examines the integration of Young functions applied to Young measures and identifies Orlicz-like structures in the space of Young measures. In particular, a convergence intermediate between the weak convergence of measures and the variational norm is determined; it serves in the completion of the Orlicz space of functions when interpreted as degenerate Young measures. Partial linear operations are defined on Young measures with respect to which the linear operations in the Orlicz space of functions are continuously embedded in the space of Young measures. This leads to a definition of convexity-type structures in the space of Young measures via a limiting procedure. These structures enable applications of Young functions arguments to Young measures. Applications to optimal control and to well posedness of minimization in function spaces with respect to convex functions are provided.     

Compactness of Bounded Quasientropy Solutions to the System of Equations of an Isothermal Gas

The notion is introduced of quasientropy solution to the equations of an isothermal gas in Lagrangian variables. Compactness of bounded quasientropy solutions is established. Reduction of Young measures to Dirac measures bases on group analysis for the entropy wave equation.     

THE APPLICATION OF THE KINETIC FORMULATION TO A SYSTEM OF QUADRATIC FLUX

In this article, the author uses the compensated compactness method coupled with some basic ideas of the kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor to give a refined proof for the existence of global entropy solutions to a system of quadratic flux. The fire-new method of reduction of Young measures is a pith of this work.     

A variational approach to a shape design problem for the wave equation

The problem of determining the optimal damping set for the stabilization of the wave equation may be not well-posed. By means of a vector variational reformulation and use of gradient Young measures, we present a general methodology to relax this kind of problems. From the optimal Young measure associated with the relaxed problem, we obtain information concerning minimizing sequences for the original problem as well as continuity properties of the relaxed cost function. To cite this article: A. Münch et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

A local proof for the characterization of Young measures generated by sequences in BV

This work presents a new proof of the recent characterization theorem for generalized Young measures generated by sequences in BV by Kristensen and Rindler (2010) [14]. The present argument is based on a localization technique together with a local Hahn–Banach argument in novel function spaces combined with an application of Alberti's Rank-One Theorem. This strategy avoids employing a relaxation theorem as in the previously known proof, and the new tools introduced in its course should prove useful in other contexts as well. In particular, we introduce “homogeneous” Young measures, separately at regular and singular points, which exhibit rather different behavior than the classical homogeneous Young measures. As an application, we show how for BV-Young measures with an “atomic” part one can find a generating sequence respecting this structure.     

Convergence of a monotonisation procedure for a non-monotone quasi-static model in poroplasticity

Existence theory to quasi-static initial-boundary value problem of poroplasticity is studied. The classical quasi-static Biot model for soil consolidation coupled with a nonlinear system of ordinary differential equations is considered. This article presents a convergence result for the coercive and monotone approximations to solution of the original non-coercive and non-monotone problem of poroplasticity such that the inelastic constitutive equation is satisfied in the sense of Young measures.     

A new approach to young measure theory, relaxation and convergence in energy

The main idea of this paper is to reduce analysis of behavior of integral functionals along weakly convergent sequences to operations with Young measures generated by these sequences. We show that Young measures can be characterized as measurable functions with values in a special compact metric space and that these functions have a spectrum of properties sufficiently broad to realize this idea.These new observations allow us to give simplified proofs of the results of gradient Young measure theory and to use them for deriving the results on relaxation and convergence in energy under optimal assumptions on integrands.We think that this work helps to clarify role of Young measures.     

Approximation theory for generalized young measures

Young measures can be understood as certain linear continuous functionals on a space of Caratheodory integrands, which gives a basis for their various generalizations. The (generalized) Young measures can be approximated by several techniques classified here according to the shape (convex/nonconvex) and dimensionality (finite/infinite) of the resulting set of all (semi)discretized generalized Young measures. A general theory for convex approximations is developed here and illustrative applications to a relaxed optimization problem are given to compare various techniques.     

Homogenization of linear and nonlinear transport equations

We study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields. Two types of homogenized equations are derived. For general n-dimensional linear and nonlinear problems, we derive homogenized equations by introducing additional independent variables to represent the small scales. For the two-dimensional linear transport equations, we derive effective equations for the averaged quantities. Such equations take the form of either a degenerate non-local diffusion equation with memory or a higher order hyperbolic equation. To study the nonlinear transport equations we introduce the concept of two-scale Young measure and extend DiPerna's method to prove that it reduces to a family of Dirac measures.     

Entropies and weak solutions of the compressible isentropic Euler equations

In this Note, we study the system of isentropic Euler equations for compressible fluids, with a general equation of state. We establish the existence of the fundamental kernel that generates the family of weak entropies, and study its singularities. The kernel is the solution of an equation of Euler-Poisson-Darboux type, and its partial derivative with respect to the density variable tends to a Dirac measure as the density approaches zero. We prove a new reduction theorem for the Young measures associated with the compressible Euler system. From these results, we deduce the existence, compactness, and asymptotic decay of measurable and bounded entropy solutions.     

On the Existence of Global Oscillation Waves for a Class of 3x3 Semilinear Hyperbolic Equations

ln this paper, we prove the global existence of oscillation waves for a class of 3 × 3 semilinear hyperbolic equations by applying the Young measures and two-scale Young measures which are associated with the solution sequence of the system.     

Fast Reaction Limit with Nonmonotone Reaction Function

We analyse the fast reaction limit in the reaction-diffusion system with nonmonotone reaction function and one nondiffusing component. As speed of reaction tends to infinity, the concentration of the nondiffusing component exhibits fast oscillations. We identify precisely its Young measure which, as a by-product, proves strong convergence of the diffusing component, a result that is not obvious from a priori estimates. Our work is based on an analysis of regularization for forward-backward parabolic equations by Plotnikov. We rewrite his ideas in terms of kinetic functions which clarifies the method, brings new insights, relaxes assumptions on model functions, and provides a weak formulation for the evolution of the Young measure. © 2022 Wiley Periodicals LLC.     

Numerical approximation of young measuresin non-convex variational problems

Summary. In non-convex optimisation problems, in particular in non-convex variational problems, there usually does not exist any classical solution but only generalised solutions which involve Young measures. In this paper, first a suitable relaxation and approximation theory is developed together with optimality conditions, and then an adaptive scheme is proposed for the efficient numerical treatment. The Young measures solving the approximate problems are usually composed only from a few atoms. This is the main argument our effective active-set type algorithm is based on. The support of those atoms is estimated from the Weierstrass maximum principle which involves a Hamiltonian whose good guess is obtained by a multilevel technique. Numerical experiments are performed in a one-dimensional variational problem and support efficiency of the algorithm. Received November 26, 1997 / Published online September 24, 1999     

Characterization of Two-Scale Gradient Young Measures and Application to Homogenization

This work is devoted to the study of two-scale gradient Young measures naturally arising in nonlinear elasticity homogenization problems. Precisely, a characterization of this class of measures is derived and an integral representation formula for homogenized energies, whose integrands satisfy very weak regularity assumptions, is obtained in terms of two-scale gradient Young measures.     

Weak solutions to a nonlinear variational wave equation and some related problems

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L ^p Young measure theory and related compactness results, in the first section. Then we use the L ^p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.     

From discrete to continuum: A Young measure approach

The passage from atomistic to continuum models is usually done via G\Gamma-convergence with respect to the weak topology of some Sobolev space; the obtained continuum energy, in a one dimensional model, is then convex. These kind of results are not optimal for problems related to materials which may undergo to phase transitions. We present here a new simple way for dealing with these problems. Our method consists in rewriting the discrete energy in terms of particular measures and taking the G\Gamma-limit with respect to the weak * convergence of measures. The continuum energy arising from a linear chain of discrete mass points interacting with only the nearest neighbours turns out to be written in terms of Young measures. While, if the discrete mass points interact not only with the nearest neighbours but also with the second nearest neighbours we obtain a continuum problem in which appears a ``multiple Young measure" representing multiple levels of interaction. In this way we obtain a novel continuum problem which is able to capture the ``microstructure" at two different levels.     

On a diffusion‐kinetic equation arising in extended kinetic theory

This paper is devoted to the study of solvability of an evolution equation obtained as a hydrodynamic limit of a linear Boltzmann equation with elastic and inelastic scattering terms. An interesting feature of this equation is that it combines the diffusion operator and a singular operator of a kinetic type. The study is carried out in the L₁ space which is natural from the physical point of view and allows to apply modifications of methods of the kinetic theory. Copyright © 2000 John Wiley & Sons, Ltd.     

Dissipative homoclinic loops of two‐dimensional maps and strange attractors with one direction of instability

We prove that when subjected to periodic forcing of the form certain two‐dimensional vector fields with dissipative homoclinic loops generate strange attractors with Sinai‐Ruelle‐Bowen measures for a set of forcing parameters (μ, ρ, ω) of positive Lebesgue measure. The proof extends ideas of Afraimovich and Shilnikov and applies the recent theory of rank 1 maps developed by Wang and Young. We prove a general theorem and then apply this theorem to an explicit model: a forced Duffing equation of the form © 2011 Wiley Periodicals, Inc.