Entropy Solutions Research Articles

Effects of A-site configuration entropy on the dielectric properties in tetragonal tungsten bronze ceramics

A series of Sr4.5Ca0.5MTi3Nb7O30 [M = La0.5Nd0.5 (LN), La1/3Nd1/3Sm1/3 (LNS) and La0.25Nd0.25Sm0.25Eu0.25 (LNSE)] ceramics with different A-site configuration entropy were prepared, and the effects of the A-site configuration entropy on the dielectric and ferroelectric properties were investigated. Due to the combined effect of high entropy and solid solution, the best dielectric performance was achieved in the LNS composition, where the dielectric constant remained around 650 in the temperature range of 269–515 K with a change rate of less than 15%. Under the DC high voltage field up to 67 kV/cm, each component's polarization characteristics and dielectric constants obtained good electric field stability and temperature stability under a high electric field.

Global existence of spherically symmetric solutions for isothermal Euler‐Poisson system outside a ball

In this paper, we consider an isothermal Euler‐Poisson system with self‐gravitational force, modeling a compact star such as a strange quark star. We prove that there exists a global entropy solution with spherical symmetry outside a ball, through the fractional Lax‐Friedrichs scheme and the theory of compensated compactness.

Navigating the Complex Landscape of Shock Filter Cahn–Hilliard Equation: From Regularized to Entropy Solutions

Navigating the Complex Landscape of Shock Filter Cahn–Hilliard Equation: From Regularized to Entropy Solutions

Degenerate Parabolic Equation with Zero-flux Boundary Condition and its Approximations

We study a degenerate parabolic-hyperbolic equation with zero-flux boundary condition. The aim of this paper is to prove convergence of numerical approximate solutions towards the unique entropy solution. We propose an implicit finite volume scheme on admissible mesh. We establish fundamental estimates and prove that the approximate solution converge towards an entropy-process solution. Contrarily to the case of Dirichlet condition, in zero-flux problem unnatural boundary regularity of the flux is required to establish that entropyprocess solution is the unique entropy solution. In the study of well-posedness of the problem, tools of nonlinear semigroup theory (stationary, mild and integral solutions) were used in order to overcome this difficulty. Indeed, in some situations including the one-dimensional setting, solutions of the stationary problem enjoy additional boundary regularity. Here, similar arguments are developed based on the new notion of integral-process solution that we introduce for this purpose.

Well-posedness results to parabolic problems involving (p(x),q(x))-growth structure with L1-data

We investigate the well-posedness (existence and uniqueness) of solutions to nonlinear parabolic problems with variable exponents and irregular data. Firstly, we establish the existence and uniqueness of weak solutions to the studied model when the data are regular enough. Secondly, we assume that the data belongs only to L1 and we prove the existence and uniqueness of both renormalized and entropy solutions. Finally, we demonstrate the equivalence between the renormalized and entropy notion of solutions to the considered problems. Our approach is based essentially on the use of truncation methods and involves some new technical estimates.

Semi-discrete Lagrangian-Eulerian approach based on the weak asymptotic method for nonlocal conservation laws in several dimensions

In this work, we have expanded upon the (local) semi-discrete Lagrangian-Eulerian method initially introduced in Abreu et al. (2022) to approximate a specific class of multi-dimensional scalar conservation laws with nonlocal flux, referred to as the nonlocal model: ∂tρ(t,x)+∑i=1d∂xi(Vi[W[ρ,ω](t,x)]Fi(ρ(t,x)))=0,(t,x)∈(0,T)×Rd. For completeness, we analyze the convergence of this method using the weak asymptotic approach introduced in Abreu et al. (2016), with significant results extended to the multidimensional nonlocal case. While there are indeed other important techniques available that can be utilized to prove the convergence of the numerical scheme, the choice of this particular technique (weak asymptotic analysis) is quite natural. This is primarily due to its suitability for dealing with the Lagrangian-Eulerian schemes proposed in this paper. Essentially, the weak asymptotic method generates a family of approximate solutions satisfying the following properties: 1) The family of approximate functions is uniformly bounded in the space L1(Rd)∩L∞(Rd). 2) The family is dominated by a suitable temporal and spatial modulus of continuity. These properties allow us to employ the L1-compactness argument to extract a convergent subsequence. We demonstrate that the limit function is a weak entropy solution of Eq. (1). Finally, we present a section of numerical examples to illustrate our results. Finally, we have examined examples discussed in Aggarwal et al. (2015) and Keimer et al. (2018). In the context of the family of two-dimensional nonlocal Burgers equations, we provide numerical results for a nonlocal impact of the form ωη∗ρ, where η=0.1.

Hyperbolic Balance Laws: Interplay between Scales and Randomness

Hyperbolic balance laws are fundamental in the mathematical modeling of transport-dominated processes in natural, socio-economic and engineering sciences. The aim of the workshop was to discuss open questions in the area of nonlinear hyperbolic conservation and balance laws. We have focused on a delicate interplay between scale hierarchies and random/stochastic effects and discuss them from analytical, numerical and modeling point of view. This leads to questions of admissibility criteria connecting to ill-posedness of weak entropy solutions, hyperbolic problems with non-local terms, mean field theory, multiscale and structure preserving numerical methods, random solutions and uncertainty quantification methods, as well as data-based methods.

Entropy solutions of elliptic equation from two phase problems

The Dirichlet problem of elliptic equation from two phase problems, b(u)−div(|∇u)|p(x)−2∇u+μ(x)|∇u|q(x)−2∇u)=f(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$b(u)-\\operatorname{div}\\left (\\left |\ abla u)\\right |^{p(x) - 2} \ abla u+\\mu (x)\\left |\ abla u\\right |^{q(x) - 2}\ abla u\\right )=f(x)$\\end{document} in Ω, u=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u=0$\\end{document} on ∂Ω is considered, where Ω is a bounded domain with a smooth boundary in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{N}$\\end{document}, b is a continuous and nondecreasing function. By the theory of the weighted variable Sobolev space, the existence and uniqueness of an entropy solution for L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{1}$\\end{document}-data f are proved.

GRP – A direct Godunov extension

The classical paper [40] by S. K. Godunov had a revolutionary effect on the field of numerical simulations of compressible fluid flows. Yet, for some twenty years after its publication, it was mostly used by the mathematical community in the Soviet Union. The seminal paper of van Leer [85] has inaugurated the period of universal interest in high-resolution extensions of Godunov's scheme. The first step to take was to modify the (locally) self-similar solution to the Riemann Problem (at discontinuities) by allowing piecewise-polynomial (rather than piecewise-constant) initial data. The GRP (Generalized Riemann Problem) analysis [6] provided analytical solutions (for piecewise-linear data) that could be readily implemented in a high-resolution robust code. This review paper focuses on the evolution of some mathematical aspects of this method. It addresses the three concepts of consistency, stability and convergence in the context of compact finite difference (or finite volume) schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”, a common viewpoint in relevant physical conservation laws. The first significant observation is that under very mild conditions a weak solution is indeed a solution to the balance law (obtained by a formal application of the Gauss-Green formula). Since high-resolution schemes require the computation of several quantities per mesh cell (e.g., slopes), the notion of “consistency” must be extended to this framework. A combination of consistency hypothesis with stability of the scheme leads to a suitable convergence theorem, generalizing the classical convergence theorem of Lax and Wendroff [52]. Finally, the limit functions are shown to be entropy solutions by using a notion of “Godunov compatibility”, which serves as a substitute to the entropy condition.

Anisotropic obstacle Neumann problems in weighted Sobolev spaces and variable exponent

Abstract In this paper we focus on a certain class of anisotropic obstacle problems governed by a Leray–Lions operator. This problem is subject to homogeneous Neumann boundary conditions. By applying truncation techniques and the monotonicity method, we establish the existence of entropy solutions for the problem studied in the framework of anisotropic weighted Sobolev spaces with variable exponent.

Entropy solutions to the fully nonlocal diffusion equations

AbstractWe consider the fully nonlocal diffusion equations with nonnegative ‐data. Based on the approximation and energy methods, we prove the existence and uniqueness of nonnegative entropy solutions for such problems. In particular, our results are valid for the time‐space fractional Laplacian equations.

Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems

We are concerned with the Prandtl-Meyer reflection configurations of unsteady global solutions for supersonic flow impinging upon a symmetric solid wedge. Prandtl (1936) first employed the shock polar analysis to show that there are two possible steady configurations: the steady weak shock solution and the steady strong shock solution, when a steady supersonic flow impinges upon the solid wedge – the half-angle of which is less than a critical angle (i.e., the detachment angle), and then conjectured that the steady weak shock solution is physically admissible since it is the one observed experimentally. The fundamental issue of whether one or both of the steady weak and strong shocks are physically admissible has been vigorously debated over the past eight decades and has not yet been settled in a definitive manner. On the other hand, the Prandtl-Meyer reflection configurations are core configurations in the structure of global entropy solutions of the two-dimensional Riemann problem, while the Riemann solutions themselves are local building blocks and determine local structures, global attractors, and large-time asymptotic states of general entropy solutions of multidimensional hyperbolic systems of conservation laws. In this sense, we have to understand the reflection configurations in order to understand fully the global entropy solutions of two-dimensional hyperbolic systems of conservation laws, including the admissibility issue for the entropy solutions. In this monograph, we address this longstanding open issue and present our analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. To achieve these, we first reformulate the problem as a free boundary problem involving transonic shocks and then obtain appropriate monotonicity properties and uniform a priori estimates for admissible solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory for all the physical parameters up to the detachment angle.

Exploring climate-change impacts on streamflow and hydropower potential: insights from CMIP6 multi-GCM analysis

ABSTRACT The generation of hydropower is profoundly influenced by shifts in streamflow patterns induced by climate change. This research examines changes in streamflow and the potential surge in hydropower generation over a span of 35 years (2015–2050) at the Bhakra Dam site within the Upper Sutlej River basin. Employing a deep learning methodology, particularly the long short-term memory (LSTM) model, in conjunction with Coupled Model Intercomparison Project (CMIP) 6 multi-global climate model (GCM), facilitates a thorough analysis of these dynamics. Six out of 14 bias-corrected statistically downscaled datasets (0.25° × 0.25° grid resolution) from CMIP6 multi-GCM were selected based on entropy and combined compromise solution techniques. This innovative approach is utilized to assess streamflows and project potential increases in hydropower at the Bhakra Dam site under shared socioeconomic pathways (SSP) scenarios, specifically SSP245 and SSP585. The results indicate a maximum increase of approximately 15% and 17% in mean monthly streamflow under SSP245 and SSP585, respectively. Moreover, dependability flows calculated at Q50, Q75, and Q90 show respective rises of 13%, 16%, and 17% under SSP245 and 21%, 17%, and 18% under SSP585. The projected hydropower potential exhibits an increase of up to 15.9% and 17.3% under SSP245 and SSP585, respectively.

Global solutions for a hyperbolic conservation system of three equations

In this short paper, we are concerned with a mathematical model, which is the variant of non-isentropic system of polytropic gas. The main contribution is to prove the a-priori L∞ estimates for the viscosity approximation solutions and to obtain a new application, on a large hyperbolic system of three equations, of the global existence framework of weak solutions in the paper “Existence of Global Entropy Solutions to a Nonstrictly Hyperbolic System” (Lu, 2005).

Delta shock wave for ternary nonlinear chromatography equations as self-similar viscosity limit

In this paper, we investigate the existence and stability of the delta shock wave in ternary chromatography equations by the self-similar viscosity vanishing approach. Considering the appropriate initial values, we prove the existence of the self-similar solution for the corresponding Riemann problem of the ternary chromatography viscous equations. Furthermore, we rigorously demonstrate that the delta shock wave is the weak star limit of the self-similar solution as viscosity tends to disappear. The result implies that the structure of the delta shock wave is stable under the self-similar viscosity perturbation, which guarantees that the delta shock wave is a unique entropy solution. In addition, we present numerical simulations in agreement with the theoretical analysis.

A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws

We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre’s hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel–Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.

Hydrodynamics for Asymmetric Simple Exclusion on a Finite Segment with Glauber-Type Source

We consider an open interacting particle system on a finite lattice. The particles perform asymmetric simple exclusion and are randomly created or destroyed at all sites, with rates that grow rapidly near the boundaries. We study the hydrodynamic limit for the particle density at the hyperbolic space-time scale and obtain the entropy solution to a boundary-driven quasilinear conservation law with a source term. Different from the usual boundary conditions introduced in Bardos et al (Commun Partial Differ Equ 4(9):1017–1034, https://doi.org/10.1080/03605307908820117, 1979) and Otto (C R Acad Sci Paris 322(1):729–734, 1996), discontinuity (boundary layer) does not formulate at the boundaries due to the strong relaxation scheme.

Hypersonic similarity for steady compressible full Euler flows over two-dimensional Lipschitz wedges

We establish the optimal convergence rate to the hypersonic similarity law, which is also called the Mach number independence principle, for steady compressible full Euler flows over two-dimensional slender Lipschitz wedges. Mathematically, it can be formulated as the comparison of the entropy solutions in BV∩L1 between the two initial-boundary value problems for the compressible full Euler equations with parameter τ>0 and the hypersonic small-disturbance equations (the scaled compressible full Euler equations with parameter τ=0) with curved characteristic boundaries. We establish the L1–convergence estimate of these two solutions with the optimal convergence rate, which justifies the Van Dyke's similarity theory rigorously for the compressible full Euler flows. This is the first mathematical result on the comparison of two solutions of the compressible Euler equations with characteristic boundary conditions. To achieve this, we first employ the special structures of the two systems and establish the global existence and the L1–stability of the entropy solutions via the wave-front tracking scheme under the smallness assumption on the total variation of both the initial data and the tangential slope function of the wedge boundary. Based on the L1–stability properties of the approximate solutions to the scaled equations with parameter τ>0, a uniform Lipschtiz continuous map with respect to the initial data and the wedge boundary is obtained, which is the first time for the characteristic boundary conditions. Next, we compare the solutions given by the Riemann solvers of the two systems by taking the boundary perturbations into account case by case. Then, for a given fixed hypersonic similarity parameter, as the Mach number tends to infinity, by employing the Lipschitz continuous properties of the map, we establish the desired L1–convergence estimate with the optimal convergence rate. Finally, we show the optimality of the convergence rate by investigating a special solution.

Inviscid Burgers as a degenerate elliptic problem

We demonstrate the feasibility of a scheme to obtain approximate weak solutions to the (inviscid) Burgers equation in conservation and Hamilton-Jacobi form, treated as degenerate elliptic problems. We show different variants recover non-unique weak solutions as appropriate, and also specific constructive approaches to recover the corresponding entropy solutions.

Formation and local stability of a two-dimensional Prandtl boundary layer system in fluid dynamics

By the linearized method, Oleǐnik-Samokhin had constructed solutions to distinguish between two cases of the behavior of the fluid at the initial stage of its motion past the surface in Oleinǐk and Samokhin (1999). In this paper, we consider the development of the boundary layer about a body that gradually starts to move in a resting fluid. In the analytical frame, the formation of the layer shows that, when the layer starts to move, the local solutions of the Prandtl boundary layer system is positive. By the Crocco transformation, the Prandtl boundary layer system reduces a degenerate parabolic equation with a nonlinear boundary value condition. Then by the reciprocal transformation, we can obtain a divergence type parabolic equation. We quote a new kind of BV entropy solution matching up with this divergence type parabolic equation when t≥t0, where t0 is a small enough positive constant. By imposing some close connections between the velocity of the outflow and the geometric characteristic of the spatial domain, using Kružkov’s bi-variables method, the local stability of BV entropy solutions is proved. Thus, by the Crocco inverse transformation, in the analytical frame, we obtain the existence and the uniqueness of the global solution of the Prandtl boundary layer system for a regular spatial domain.