Measure Valued Solutions of the 2D Keller–Segel System

We develop a theory of global measure-valued solutions for the classical Keller–Segel model. These solutions are obtained considering the limit of solutions of a regularized problem. We also prove that different regularizations yield different limit measures in the case in which classical solutions of the Keller–Segel system are not globally defined in time.     

Existence and Strong Pre-compactness Properties for Entropy Solutions of a First-Order Quasilinear Equation with Discontinuous Flux

Sequences of entropy solutions of a non-degenerate first-order quasilinear equation are shown to be strongly pre-compact in the general case of a Caratheodory flux vector. Existence of the weak and entropy solution to the Cauchy problem for such an evolutionary equation is also established. The proofs are based on the general localization principle for H-measures corresponding to sequences of measure-valued functions.     

A Variational Approximation Scheme for¶Three-Dimensional Elastodynamics¶with Polyconvex Energy

We construct a variational approximation scheme for the equations of three-dimensional elastodynamics with polyconvex stored energy. The scheme is motivated by some recently discovered geometric identities (Qin [18]) for the null Lagrangians (the determinant and cofactor matrix), and by an associated embedding of the equations of elastodynamics into an enlarged system which is endowed with a convex entropy. The scheme decreases the energy, and its solvability is reduced to the solution of a constrained convex minimization problem. We prove that the approximating process admits regular weak solutions, which in the limit produce a measure-valued solution for polyconvex elastodynamics that satisfies the classical weak form of the geometric identities. This latter property is related to the weak continuity properties of minors of Jacobian matrices, here exploited in a time-dependent setting. Accepted November 18, 2000?Published online April 23, 2001     

The Continuous Coagulation-Fragmentation¶Equations with Diffusion

Existence of global weak solutions to the continuous coagulation-fragmentation equations with diffusion is investigated when the kinetic coefficients satisfy a detailed balance condition or the coagulation coefficient enjoys a monotonicity condition. Our approach relies on weak and strong compactness methods in L ¹ in the spirit of the DiPerna-Lions theory for the Boltzmann equation. Under the detailed balance condition the large-time behaviour is also studied.     

Weak–Strong Uniqueness of Dissipative Measure-Valued Solutions for Polyconvex Elastodynamics

For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of a dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data, when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging.     

Initial value problem for high dimensional dynamic systems

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Functional and measure-valued solutions of the euler equations for flows of incompressible fluids

We consider the notion of a functional solution of the Euler equations for incompressible fluid flows. We show that a functional solution can be constructed under very weak a priori estimates on approximate solution sequences of the equation; an estimate uniform in L _loc ¹ together with weak consistency with the equation is sufficient to construct a solution. We prove that if we have an estimate uniform in L _loc ² available for the approximate solution sequence, then the structured functional solution just described becomes a measure-valued solution in the sense of DiPerna & Majda. We also show that a functional solution can be obtained from a measure-valued solution. We give an example showing that a much higher concentration of energy than in the case of measure-valued solutions is allowed by the approximation procedure of a functional solution.     

Discontinuous deformation gradients in the finite twisting of an incompressible elastic tube

Summary It is known that the type of the system of partial differential equations governing finite elastostatics can change from elliptic to non-elliptic at sufficiently large deformations for certain materials. This introduces the possibility that the elastostatic field may exhibit certain discontinuities. Some aspects of the general theory associated with these issues were examined in a recent series of studies by Knowles and Sternberg. In this paper we illustrate the occurrence of elastostatic fields with discontinuous deformation gradients in a physical problem. The body is assumed to be composed of a material which belongs to a particular class of isotropic, incompressible, elastic materials which allow for a loss of ellipticity. It is shown that no solution which is smooth in the classical sense exists to this problem for certain ranges of the applied loading. Next, we admit solutions involving elastostatic shocks into the discussion and find that the problem may then be solved completely. When this is done, however, there results a lack of uniqueness of solutions to the boundary-value problem. In order to resolve this non-uniqueness, the dissipativity and stability of the solutions are investigated.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington D.C.     

On the accuracy of a nonlinear finite volume method for the solution of diffusion problems using different interpolations strategies

In this paper, we consider a nonlinear finite volume method to solve the steady‐state diffusion equation in nonhomogeneous and non‐isotropic media. The method is nonlinear even if the original problem is linear. In its original form, the scheme is monotone, because the coefficient matrix is monotone under certain assumptions and, as a consequence, whenever the analytic operator demands, it preserves the positivity of numerical solutions. On the other hand, the scheme is unable to reproduce piecewise linear solutions exactly. In order to recover this interesting feature, we use two different interpolation strategies. In this case, even though we are unable to prove monotonicity, we show some numerical evidences that the combined method has an improved behavior, producing second order accurate solutions, even for nonhomogeneous and strongly anisotropic media. Copyright © 2013 John Wiley & Sons, Ltd.     

Paradox solution on elastic wedge dissimilar materials

According to the Hellinger-Reissner variational principle and introducing proper transformation of variables , the problem on elastic wedge dissimilar materials can be led to Hamiltonian system, so the solution of the problem can be got by employing the separation of variables method and symplectic eigenfunction expansion under symplectic space, which consists of original variables and their dual variables . The eigenvalue - 1 is a special one of all symplectic eigenvalue for Hamiltonian system in polar coordinate . In general, the eigenvalue - 1 is a single eigenvalue, and the classical solution of an elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is got directly by solving the eigenfunction vector for eigenvalue - 1. But the eigenvalue - 1 becomes a double eigenvalue when the vertex angles and modulus of the materials satisfy certain definite relationships and the classical solution for the stress distribution becomes infinite at this moment, that is, the para     

Optimal Control of Turbulent Flow as Measure Solutions

Abstract

We use the concept of relaxed or measure-valued solutions for control problems of turbulent flow related to Navicr-Stokes equation. Sufficient conditions guaranteeing the existence of measure solutions arc presented. Results on existence of optimal controls for Blow up time and Bolza problems for such systems are also presented. New results on relaxed necessary conditions of optimality are proved. Further it is shown that the relaxed necessary conditions reduce to classical Pontryagin type necessary conditions if measure solutions degenerate into Dirac structure. The paper is concluded with an algorithm based on the new necessary conditions for computing optimal controls.     

Measure-valued solutions of scalar conservation laws with boundary conditions

We define a solution concept for measure-valued solutions to scalar conservation laws with initial conditions and boundary conditions and prove a uniqueness theorem for such solutions. This result may be used to prove convergence, towards the unique solution, for approximate solutions which are uniformly bounded in L , weakly consistent with certain entropy inequalities and strongly consistent with the initial condition, i.e. without using derivative estimates. As an example convergence of a finite element method is demonstrated.     

Young Measures Generated by Ideal Incompressible Fluid Flows

In their seminal paper, DiPerna and Majda (Commun Math Phys 108(4):667–689, 1987) introduced the notion of a measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a very large, arguably too large, set of weak solutions of the incompressible Euler equations.     

Convergence of the phase-field equations to the mullins-sekerka problem with kinetic undercooling

I prove that the solutions of the phase-field equations, on a subsequence, converge to a weak solution of the Mullins-Sekerka problem with kinetic undercooling. The method is based on energy estimates, a monotonicity formula, and the equipartition of the energy at each time. I also show that for almost all t, the limiting interface is (d – 1)-rectifiable with a square-integrable mean-curvature vector.Dedicated to Mort Gurtin on the occasion of his sixtieth birthday     

Similarity method for imploding strong shocks in a non-ideal relaxing gas

Self-similar solutions arise naturally as special solutions of system of partial differential equations (PDEs) from dimensional analysis and, more generally, from the invariance of system of PDEs under scaling of variables. Usually, such solutions do not globally satisfy imposed boundary conditions. However, through delicate analysis, one can often show that a self-similar solution holds asymptotically in certain identified domains. In the present paper, it is shown that self-similar phenomena can be studied through use of many ideas arising in the study of dynamical systems. In particular, there is a discussion of the role of symmetries in the context of self-similar dynamics. We use the method of Lie group invariance to determine the class of self-similar solutions to a problem involving plane and radially symmetric flows of a relaxing non-ideal gas involving strong shocks. The ambient gas ahead of the shock is considered to be homogeneous. The method yields a general form of the relaxation rate for which the self-similar solutions are admitted. The arbitrary constants, occurring in the expressions for the generators of the local Lie group of transformations, give rise to different cases of possible solutions with a power law, exponential or logarithmic shock paths. In contrast to situations without relaxation, the inclusion of relaxation effects imply constraint conditions. A particular case of the collapse of an imploding shock is worked out in detail for radially symmetric flows. Numerical calculations have been performed to determine the values of the self-similarity exponent and the profile of the flow variables behind the shock. All computations are performed using the computation package Mathematica.     

Nonlinear Elliptic–Parabolic Problems

We introduce a notion of viscosity solutions for a general class of elliptic–parabolic phase transition problems. These include the Richards equation, which is a classical model in filtration theory. Existence and uniqueness results are proved via the comparison principle. In particular, we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data. These results are new, even in the linear case, where we also show that viscosity solutions coincide with the regular weak solutions introduced in Alt and Luckhaus (Math Z 183:311–341, 1983).     

Motion of a Graph by <Emphasis Type="Italic">R</Emphasis>-Curvature

We show the existence of weak solutions to the partial differential equation which describes the motion by R-curvature in R ^d , by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the partial differential equation are viscosity solutions and give the uniqueness result on both weak and viscosity solutions.     

Designing Freeform Lenses for Intensity and Phase Control of Coherent Light with Help from Geometry and Mass Transport

We study here the problem of determining a system of two refractive interfaces transforming a plane wavefront of a given shape and radiation intensity into a coherent output plane wavefront with prescribed output position, shape and intensity. Such interfaces can be refracting surfaces of two different lenses or of one lens. In geometrical optics approximation, the analytic formulation of this problem in both cases requires construction of maps with controlled Jacobian. Though this Jacobian can be expressed as a second order partial differential equation of Monge-Ampère type for a scalar function defining one of the refracting surfaces, its analysis is not straightforward. In this paper we use a geometric approach for reformulating the problem in certain associated measures and defining weak solutions. Existence and uniqueness of weak solutions in Lipschitz classes for both cases are established by variational methods. Our results show, in particular, that two types of interfaces exist in each case for the same data: one of these types always consists of two interfaces, one of which is concave or convex and the second convex or concave, while the interfaces of the second type may be neither convex nor concave. The availability of a design with convex/concave lenses is particularly important for fabrication. The truly geometric nature of this problem permits its statement and investigation in \mathbb R^N+1, N \geqq 1{\mathbb {R}^{N+1},\, N \geqq 1} .     

On Phase-Separation Models: Asymptotics and Qualitative Properties

In this paper we study bound state solutions of a class of two-component nonlinear elliptic systems with a large parameter tending to infinity. The large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains divided by an interface. To obtain the profile of bound state solutions near the interface, we prove the uniform Lipschitz continuity of bound state solutions when the spatial dimension is N = 1. Furthermore, we show that the limiting nonlinear elliptic system that arises has unbounded solutions with symmetry and monotonicity. These unbounded solutions are useful for rigorously deriving the asymptotic expansion of the minimizing energy which is consistent with the hypothesis of Du and Zhang (Discontin Dynam Sys, 2012). When the spatial dimension is N = 2, we establish the De Giorgi type conjecture for the blow-up nonlinear elliptic system under suitable conditions at infinity on bound state solutions. These results naturally lead us to formulate De Giorgi type conjectures for these types of systems in higher dimensions.     

Generalized solutions of beams with jump discontinuities on elastic foundations

Summary  The bending solutions of the Euler–Bernoulli and the Timoshenko beams with material and geometric discontinuities are developed in the space of generalized functions. Unlike the classical solutions of discontinuous beams, which are expressed in terms of multiple expressions that are valid in different regions of the beam, the generalized solutions are expressed in terms of a single expression on the entire domain. It is shown that the boundary-value problems describing the bending of beams with jump discontinuities on discontinuous elastic foundations have more compact forms in the space of generalized functions than they do in the space of classical functions. Also, fewer continuity conditions need to be satisfied if the problem is formulated in the space of generalized functions. It is demonstrated that using the theory of distributions (i.e. generalized functions) makes finding analytical solutions for this class of problems more efficient compared to the traditional methods, and, in some cases, the theory of distributions can lead to interesting qualitative results. Examples are presented to show the efficiency of using the theory of generalized functions. Received 6 June 2000; accepted for publication 24 October 2000