On regularity and rigidity of $2\times 2$ differential inclusions into nonelliptic curves

Deformations with finitely many gradients and stability of quasiconvex hulls

A new nonlinear functional for general scalar conservation laws

Conservation Based Uncertainty Propagation in Dynamic Systems by Lillian J Ratliff Dr. Pushkin Kachroo, Examination Committee Chair Professor of Electrical and Computer Engineering University of Nevada, Las Vegas Uncertainty is present in our everyday decision making process as well as our understanding of the structure of the universe. As a result an intense and mathematically rigorous study of how uncertainty propagates in the dynamic systems present in our lives is warranted and arguably necessary. In this thesis we examine existing methods for uncertainty propagation in dynamic systems and present the results of a literature survey that justifies the development of a conservation based method of uncertainty propagation. Conservation methods are physics based and physics drives our understanding of the physical world. Thus, it makes perfect sense to formulate an understanding of uncertainty propagation in terms of one of the fundamental concepts in physics: conservation. We develop that theory for a small group of dynamic systems which are fundamental. They include ordinary differential equations, finite difference equations, differential inclusions and inequalities, stochastic differential equations, and Markov chains. The study presented considers uncertainty propagation from the initial condition where the initial condition is given as a prior distribution defined within a probability structure. This probability structure is preserved in the sense of measure. The results of this study are the first steps into a generalized theory for uncertainty propagation using conservation laws. In addition, it is hoped that the iii results can be used in applications such as robust control design for everything from transportation systems to financial markets.

Sparse recovery via differential inclusions

In this paper, we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in [B. Kirchheim, S. Müller and V. Šverák, Studying Nonlinear PDE by Geometry in Matrix Space (Springer, 2003), Sec. 7], and many of its properties have already been shown in [A. Lorent and G. Peng, Null Lagrangian measures in subspaces, compensated compactness and conservation laws, Arch. Ration. Mech. Anal. 234(2) (2019) 857–910; A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156]. In particular, in [A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156], it is shown that the differential inclusion does not contain any [Formula: see text] configurations. Here, we continue that study by showing that the differential inclusion does not contain [Formula: see text] configurations.

A discrete subset of C n \mathbb C^n is said to be tame if there is an automorphism of C n \mathbb C^n taking the given discrete subset to a subset of a complex line; such tame sets are known to allow interpolation by automorphisms. We give here a fairly general sufficient condition for a discrete set to be tame. In a related direction, we show that for certain discrete sets in C n \mathbb C^n there is an injective holomorphic map from C n \mathbb C^n into itself whose image avoids an ϵ \epsilon -neighborhood of the discrete set. Among other things, this is used to show that, given any complex n n -torus and any finite set in this torus, there exist an open set containing the finite set and a locally biholomorphic map from C n \mathbb C^n into the complement of this open set.

We consider the inhomogeneous incompressible Euler equations including their local energy inequality as a differential inclusion. Providing a corresponding convex integration theorem and constructing subsolutions, we show the existence of locally dissipative Euler flows emanating from the horizontally flat Rayleigh–Taylor configuration and having a mixing zone which grows quadratically in time. For the Rayleigh–Taylor instability these are the first turbulently mixing solutions known to respect local energy dissipation, and outside the range of Atwood numbers considered in Gebhard et al. (Arch Ration Mech Anal 241(3):1243–1280, 2021), the first weakly admissible solutions in general. In the coarse grained picture the existence relies on one-dimensional subsolutions described by a family of hyperbolic conservation laws, among which one can find the optimal background profile appearing in the scale invariant bounds from Kalinin et al. (SIAM J Math Anal 56(6):7846–7865, 2024), and as we show, the optimal conservation law with respect to maximization of the total energy dissipation. Furthermore, we also show that the least action admissibility criteria from Gimperlein et al. (Arch Ration Mech Anal 249(2):22, 2025; arXiv:2503.03491, 2025) selects rather the stationary solution within our family of conservation laws.

We present an existence result for a partial differential inclusion with linear parabolic principal part and relaxed one-sided Lipschitz multivalued nonlinearity in the framework of Gelfand triples. Our study uses discretizations of the differential inclusion by a Galerkin scheme, which is compatible with a conforming finite element method, and we analyze convergence properties of the discrete solution sets.

In this paper, a variational framework is proposed for the constitutive update of thermomechanical constitutive models in the special case where their input results from quantities directly updated by hyperbolic conservation laws. Both a continuum and a consistent first order accurate discrete settings are derived. The originality of this work lies in that the constitutive update is driven by the rates of some strain measure and the internal energy density in the continuum setting, leading to a rate-type description of the local constitutive problem, and by the updated values at some discrete time of these strain measure and internal energy density in the discrete setting. These quantities are updated by the solution of a system of discrete conservation laws including the first principle of thermodynamics, ensuring that the right shock speeds will be computed. This point is of crucial importance when simulating impact on structures for instance. The proposed variational approach is illustrated for thermo-hyperelastic-viscoplastic solid media, especially using the parameterization of the flow rule direction based on pseudo-stresses proposed by Mosler & co-workers. The proposed discrete variational solver is then coupled with the second order accurate flux difference splitting finite volume method, which permits to solve the set of conservation laws. Comparisons are performed on a set of test cases with numerical solutions obtained with finite elements coupled to an explicit time-stepping and to a temperature-driven variational constitutive update. They allow to show the good behavior of the proposed approach.

The Survey and Review paper of this issue, “Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws,” by G. A. El, M. A. Hoefer, and M. Shearer, connects two well-known areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves, and uses a raft of applied mathematics tools: phase portrait analysis, multiple scales, averaging, etc. The theory of nonlinear dispersive waves has its roots in nineteenth century studies of water waves and gained an enormous prominence after the computer discovery of the soliton by Zabusky and Kruskal in 1965. This discovery led to the theory of integrable Hamiltonian partial differential equations with ramifications in many areas of mathematics, and also in mathematical physics, nonlinear optics, etc. The evolution of the theory of hyperbolic conservation laws has followed a similar pattern; while it started with Riemann's work on gas dynamics, it only came of age in the 1950s and 1960s with the contributions of Lax, Oleinik, and others. Regardless of the smoothness of the initial data, solutions of nonlinear conservation laws typically develop discontinuities and therefore have to be understood in a weak sense. To make matters more complicated, weak solutions are often not unique, and the solution with physical relevance has to be identified with the help of an entropy condition. A way to avoid these difficulties is to regularize the differential equations. Regularization is of course not only a matter of mathematical convenience: it brings to the equation “higher order” physical effects ignored in the conservation law formulation. The dissipative regularization of the inviscid Burgers equation \( u_t+(u^2/2)_x = 0\) results in the Burgers equation \( u_t+(u^2/2)_x = \epsilon u_xx\), \(\epsilon >0\). The inviscid shock that jumps from the upstream state \( u^-\) to the downstream state \(u^+\), \(u^->u^+\), is regularized to become a viscous, smooth traveling wave solution that shares the shock speed and upstream and downstream values. In situations without dissipation, regularization may arise from dispersion, as in the Korteweg--de Vries (KdV) equation \( u_t+(u^2/2)_x = \mu u_xxx\), and in that case the regularization of shock solutions leads to so-called dispersive shock waves, entities with a structure much richer than that of the dissipative traveling wave. The paper focuses on the modified KdV Burgers equation \( u_t+(u^3)_x = \epsilon u_xx+\mu u_xxx\), a universal asymptotic model where the corresponding conservation law \( u_t+(u^3)_x = 0\) has a nonconvex flow \( u^3\). As the paper shows, many remarkable phenomena appear due to the competition among nonconvexity, dissipation, and dispersion; these include undercompressive shock waves and double-wave complexes such as shock-rarefactions.

The Brio system is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics. The system has the form It was found in previous works that the standard theory of hyperbolic conservation laws does not apply to this system since the characteristic fields are not genuinely nonlinear on the set . As a consequence, certain Riemann problems have no weak solutions in the traditional Lax admissible sense.It was argued in Hayes and LeFloch (1996 Nonlinearity 9 1547–63) that in order to solve the system, singular solutions containing Dirac masses along the shock waves might have to be used. Solutions of this type were exhibited in Kalisch and Mitrović (2012 Proc. Edinburgh Math. Soc. 55 711–29) and Sarrico (2015 Russ. J. Math. Phys. 22 518–27), but uniqueness was not obtained.In the current work, we introduce a nonlinear change of variables which makes it possible to solve the Riemann problem in the framework of the standard theory of conservation laws. In addition, we develop a criterion which leads to an admissibility condition for singular solutions of the original system, and it can be shown that admissible solutions are unique in the framework developed here.

is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws. new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles. From the reviews of the 3rd edition: This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject. Evgeniy Panov, Zentralblatt MATH A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the Bible on the subject. Philippe G. LeFloch, Math. Reviews

We formulate a general theory of conservation laws and other invariants for a physical system through equivalence relations. The conservation laws are classified according to the type of equivalence relation, with group equivalence, homotopical equivalence, and other types of equivalence relations giving respective kinds of conservation laws. The stability properties in the topological (and differentiable) sense are discussed using continuous deformations with respect to control parameters. The conservation laws due to the Abelian symmetries are shown to be stable through application of well-known theorems.

We examine the two-dimensional Euler equations including the local energy (in)equality as a differential inclusion and show that the associated relaxation essentially reduces to the known relaxation for the Euler equations considered without local energy (im)balance. Concerning bounded solutions we provide a sufficient criterion for a globally dissipative subsolution to induce infinitely many globally dissipative solutions having the same initial data, pressure, and dissipation measure as the subsolution. The criterion can easily be verified in the case of a flat vortex sheet giving rise to the Kelvin--Helmholtz instability. As another application we show that there exists initial data for which associated globally dissipative solutions realize every dissipation measure from an open set in ${\mathcal C}^0(\mathbb{T}^2\times[0,T])$. In fact the set of such initial data is dense in the space of solenoidal $L^2(\mathbb{T}^2;\mathbb{R}^2)$ vector fields.

Finite volume methods are a class of discretization schemes resulting from the decomposition of a problem domain into nonoverlapping control volumes. Degrees of freedom are assigned to each control volume that determine local approximation spaces and quadratures used in the calculation of control volume surface fluxes and interior integrals. An imposition of conservation and balance law statements in each and every control volume constrains surface fluxes and results in a coupled system of equations for the unknown degrees of freedom that must be solved by a numerical method.Finite volume methods have proved highly successful in approximating the solution to a wide variety of conservation and balance laws. They are extensively used in fluid mechanics, meteorology, electromagnetics, semiconductor device simulation, materials modeling, heat transfer, models of biological processes, and many other engineering problems governed by conservation and balance laws that may be written in integral control volume form.This chapter reviews elements of the foundation and analysis of modern finite volume methods for approximating hyperbolic, elliptic, and parabolic partial differential equations. These different equations have markedly different continuous problem regularity and function spaces (e.g., , , and ) that must be adequately represented in finite‐dimensional discretizations. Particular attention is given to finite volume discretizations yielding numerical solutions that inherit properties of the underlying continuous solutions such as maximum (minimum) principles, total variation control, stability, global entropy decay, and local balance law conservation while also having favorable accuracy and convergence properties on structured and unstructured meshes.As a starting point, a review of scalar nonlinear hyperbolic conservation laws and the development of high‐order accurate schemes for discretizing them is presented. A key tool in the design and analysis of finite volume schemes suitable for discontinuity capturing is discrete maximum principle analysis. A number of mathematical and algorithmic developments used in the construction of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions. These developments include monotone fluxes, TVD discretization, positive coefficient discretization, nonoscillatory reconstruction, slope limiters, strong stability preserving time integrators, and so on. When available, theoretical results concerninga priorianda posteriorierror estimates and convergence to entropy weak solutions are given.A review of the discretization of elliptic and parabolic problems is then presented. The tools needed for the theoretical analysis of the two point flux approximation scheme for the convection diffusion equation are described. Such schemes require an orthogonality condition on the mesh in order for the numerical fluxes to be consistent. Under this condition, the scheme may be shown to be monotone. A weak formulation of the scheme is derived, which facilitates obtaining stability, convergence, and error estimate results. The discretization of anisotropic problems is then considered and a review is given of some of the numerous schemes that have been designed in recent years, along with their properties. Parabolic problems are then addressed, both in the linear and nonlinear cases.A discussion of further advanced topics is then given including the extension of the finite volume method to systems of hyperbolic conservation laws. Numerical flux functions based on an exact or approximate solution of the Riemann problem of gas dynamics are discussed. This is followed by the review of another class of numerical flux functions for symmetrizable systems of conservation laws that yield finite volume solutions with provable global decay of the total mathematical entropy for a closed entropy system, often referred to as entropy stability.Finally, a detailed review of the discretization of the steady‐state incompressible Navier–Stokes equations using the Marker‐And‐Cell (MAC) finite volume method is then presented. The MAC scheme uses a staggered mesh discretization for pressure and velocities on primal and dual control volumes. After reformulating the MAC scheme in weak form, analysis results concerning stability, weak consistency, and convergence are given.