Weak Solutions Research Articles

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.

Multiple solutions for a p ( x ) -Laplacian-like problem under Neumann boundary condition

UDC 517.9 We prove the existence of at least three weak solutions for a class of p ( x ) -Laplacian-like problems with Neumann boundary conditions by using a critical theorem of Bonanno and Marano [Appl. Anal., <strong>89</strong>, 1–10 (2010)].

Global in Time Weak Solutions to Singular Three-Dimensional Quasi-Geostrophic Systems

Global in Time Weak Solutions to Singular Three-Dimensional Quasi-Geostrophic Systems

Strongly quasilinear elliptic systems with perturbations through yong measures

We use the Galerkin method to obtain weak solutions for strongly quasilinear elliptic systems in the Sobolev space W1,p0(Ω; ℝm) of the form Gϖ(x) = f(x,ϖ,Dϖ) in Ω, with ϖ(x) = 0 on ∂Ω, where the term Gϖ(x) includes perturbations. Our approach involves the use of Young measures to obtain the desired results.

Weak, Strong and Global Solutions for DNA Model with Memory

Objective: To determine solutions of hyperbolic partial differential equations with memory using operator techniques with the corresponding estimates. The hyperbolic model associated with the Peyrard-Bishop model of DNA with memory is especially analyzed. Method: Using Gronwall inequalities and operator theory, solutions to partial differential equations are obtained after reduction to an equivalent one. Results and Discussion: The existence of three types of weak, strong and global solutions for the hyperbolic model associated with the Peyrard-Bishop model of DNA with memory has been demonstrated. Implications of the research: Apply in the first university years the application of MATLAB and Python to partial differential equations in all university careers and specialties as part of the management of the tools of Information and Communication Technology (TIC). Originality/Value: This paper also provides an alternative approach to Analysis of DNA vibration amplitudes with past-dependent memory.

Framework Integrated with Flow Cell Experiments and Simulations for Understanding Etching in Chemical Conversion Treatments

In this study, a new framework integrates simulations and flow cell experimentation to quantitatively understand the mechanism of chemical treatment reactions. Using this framework, the mechanisms of etching reactions induced by weak and strong acids were specifically investigated. A flow cell system experiment was developed for the etching experiment. Two acids (HNO3 and HF) were used, along with HNO3 without electrolytes. Average flow velocities were measured, and the molar flux of Fe2+ ions was determined by sampling the solution passing through the flow cell and measuring the iron content by using inductively coupled plasma. A concentration field simulation of the etching reaction in the flow cell was conducted. The concentration field within the boundary layer was visualized to understand the mechanism of H+ ion supply to the metal surface. In the case of weak acid solutions, H+ ions are primarily supplied by dissociation. In contrast, they were supplied by diffusion in strong acid solutions. A boundary layer formed within 100 μm from the metal surface. The experimental and simulated molar flux of Fe2+ ions were compared. The molar flux attributed to weak acid etching was more than 10 times that attributed to strong acids. The reaction rate constant of the H+ reduction reaction was evaluated through a parameter study. The influence of spectator ions on the etching process was investigated. An experiment was conducted to compare the etching of iron plates using HNO3 solutions with different concentrations of spectator ion. The results confirmed that the higher the concentration of the spectator ion, the greater the etching amount. Numerical analysis revealed that the electric field in the electric migration term acts in a direction that impedes the movement of H+ ions to the metal surface. While it is already known that electric migration inhibits electrode reactions, this study enabled its quantitative visualization and evaluation.

The optimal polynomial decay in the extensible Timoshenko system

AbstractIn this paper, we derive the equations that constitute the nonlinear mathematical model of an extensible thermoelastic Timoshenko system. The nonlinear governing equations are derived by applying the Hamilton principle to full von Kármán equations. The model takes account of the effects of extensibility, where the dissipations are entirely contributed by temperature. Based on the semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By using a resolvent criterion, developed by Borichev and Tomilov, we prove the optimality of the polynomial decay rate of the considered problem under the condition (65). Moreover, by an approach based on the Gearhart–Herbst–Prüss–Huang theorem, we show the non‐exponential stability of the same problem; but strongly stable by following a result due to Arendt–Batty. In the absence of additional mechanical dissipations, the system is often not highly stable. By adding a damping frictional function to the first equation of the nonlinear derived model with extensibility and using the multiplier method, we show that the solutions decay exponentially if Equation (85) holds.

A Godunov-type scheme for the Saint-Venant-Exner equations with a moving steady states

The Saint-Venant-Exner system is widely used in industrial codes to model the transport of bed sediments. In practice, most of the methods used for its numerical resolution suffer from significant stability problems due to the fact that they are mainly guaranteed by separation techniques which allow weak coupling between hydraulic and morphodynamic software. The search for an adequate alternative method to this problem is a focus towards which several approaches converge. Recently, many authors have proposed acceptable methods but they are not so trivial to implement in an industrial context and some do not take into account all aspects such as mobile steady states. In this work, we derive a well-balanced scheme that takes into account all stable states, to approach weak solutions of the sediment transport model in shallow waters. We demonstrate that it captures exactly all regular steady solutions with non-evanescent velocities, retains water level positivity and is able to degenerate to a classical shallow water model when the sedimentary transport flow is null. A number of numerical test cases are also presented to illustrate these properties.

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.

Global existence of non-Newtonian incompressible fluid in half space with nonhomogeneous initial-boundary data

In this study, we investigate the global existence of weak solutions of non-Newtonian incompressible fluids governed by (1.1). When u0∈Ḃp,qα−2p(R+n)∩Ḃn+22,n+221−4n+2(R+n)∩Ḃp,11+np(R+n) is given, we will find the weak solutions for the Eq. (1.1) in the function space Cb[0,∞;Ḃp,qα−2p(R+n)∩Cb(0,∞;Ḃn+221−4n+2(R+n))∩L∞(0,∞;Ẇ∞1(R+n)), n+2<p<∞,1≤q≤∞,1+n+2p<α<2. We show the existence of weak solutions in the anisotropic Besov spaces Ḃp,qα,α2(R+n×(0,∞)) (see Theorem 1.2) and we show the embedding Ḃp,qα,α2R+n×(0,∞)⊂Cb[0,∞;Ḃp,qα−2p(R+n) (see Lemma 2.8). For the global existence of solutions, we assume that the extra stress tensor S is represented by S(A)=F(A)A, where F satisfies the assumption (A). Note that S1, S2 and S3 introduced in (1.2) satisfy our assumptions.

Analysis and numerical simulation of a generalized compressible Cahn–Hilliard–Navier–Stokes model with friction effects

We propose a new generalized compressible diphasic Navier–Stokes Cahn–Hilliard model that we name G-NSCH. This new G-NSCH model takes into account important properties of diphasic compressible fluids such as possible non-matching densities and contrast in mechanical properties (viscosity, friction) between the two phases of the fluid. The model also comprises a term to account for possible exchange of mass between the two phases. Our G-NSCH system is derived rigorously and satisfies basic mechanics of fluids and thermodynamics of particles. Under some simplifying assumptions, we prove the existence of global weak solutions. We also propose a structure preserving numerical scheme based on the scalar auxiliary variable method to simulate our system and present some numerical simulations validating the properties of the numerical scheme and illustrating the solutions of the G-NSCH model.

On the modulus of continuity of solutions to nonlocal parabolic equations

AbstractA general modulus of continuity is quantified for locally bounded, local, weak solutions to nonlocal parabolic equations, under a minimal tail condition. Hölder modulus of continuity is then deduced under a slightly stronger tail condition. These regularity estimates are demonstrated under the framework of nonlocal ‐Laplacian with measurable kernels.

Remarks on the Uniqueness of Weak Solutions of the Incompressible Navier–Stokes Equations

Remarks on the Uniqueness of Weak Solutions of the Incompressible Navier–Stokes Equations

Global solution of spherically symmetric compressible Navier–Stokes equations with bounded density and density‐dependent viscosity

We consider the compressible Navier–Stokes equations with viscosities in bounded domains when the initial data are spherically symmetric, which covers the Saint‐Venant model for the motion of shallow water. First, based on the exploitation of the one‐dimensional feature of symmetric solutions, we prove the global existence of weak solutions with initial vacuum, where the upper bound of the density is obtained. Then, with more conditions imposed on the nonvacuum initial data, we obtain the global weak solution which is a strong one away from the symmetry center. The analysis allows for the possibility that a vacuum state emerges at the symmetry center; in particular, we give the uniform bound of the radius of the vacuum domain.

Weak solutions of anisotropic (and crystalline) inverse mean curvature flow as limits of p-capacitary potentials

We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in RN with no ellipticity nor smoothness requirements, in order to include the crystalline case) and on the initial data. By means of an approximation procedure introduced by Moser, our solutions are limits of anisotropic p-harmonic functions or p-capacitary functions (after a change of variable), and we get uniqueness both for the approximating solutions (i.e., uniqueness of p-capacitary functions) and the limiting ones. Our notion of weak solution still recovers variational and geometric definitions similar to those introduced by Huisken-Ilmanen, but requires to work within the broader setting of BV-functions. Despite of this, we still reach classical results like the continuity and exponential growth of perimeter, as well as outward minimizing properties of the sublevel sets. Moreover, by assuming the extra regularity given by an interior rolling ball condition (where a sliding Wulff shape plays the role of a ball), the solutions are shown to be continuous and satisfy Harnack inequalities. Finally, examples of explicit solutions are built.

Blow-up and global existence for a new class of parabolic p(x,⋅)-Kirchhoff equation involving double phase operator

In this paper, we are concerned with study solutions for double phase problems involving a Kirchhoff function, and nonlinearity with subcritical growth:{ut+M([u]p,q,as)Lp,q,asu=|u|r−2u,inU×[0,T),u(x,t)=0in∂U×[0,T),u(x,0)=u0(x),inU, where, Lp,q,ss is double phase operator, and M is a Kirchhoff function. Combining the Faedo-Galerking approximation with Gronwall's inequality, the existence of a unique weak solution is established. More interestingly, we study the global existence and finite time blow of solution when the initial energy is sub-critical and supercritical. Finally, the stabilization of the solution with positive initial energy is established based on Komornik's integral inequality.

Existence of Solutions for a Viscoelastic Plate Equation with Variable Exponents and a General Source Term

The subject of this study is a nonlinear viscoelastic plate equation with variable exponents and a general source term. Through the application of the Faedo–Galerkin approximation method and a fixed point theorem under appropriate assumptions, we proved the existence of weak solutions.

An efficient and accurate mapping method for elliptic equations in irregular annular domains

In this paper, we introduce a coordinate transformation, which transforms the irregular annular domain to a unit disk. We present its basic properties. As examples, we consider Poisson type equation and Cauchy–Navier elastic equations with variable coefficients in two-dimensional irregular annular domains, and prove the existence and uniqueness of weak solutions. We also construct the mixed Fourier–Legendre spectral schemes, and derive the optimal convergence of numerical solutions under the H1-norm. The numerical results indicate that the suggested method achieves high-order accuracy.

Double Phase Variable Exponent Problems with Nonlinear Matrices Diffusion

This work tackles a class of double phase elliptic problems with variable exponents and matrices diffusion. Under suitable assumptions on the data, we use critical point theory to establish both the existence and uniqueness of weak solutions to the double phase problem under consideration.

Existence of solutions for a hyperbolic Kirchhoff-type problem with a volume constraint

In this study, we investigate the existence of weak solutions for a hyperbolic Kirchhoff-type problem constrained by volume. Our research is motivated by the complex nature of such problems, which involve non-local terms arising from the Kirchhoff term and the volume constraint. These non-local terms introduce significant challenges, complicating the application of traditional analytical methods and necessitating the development of innovative approaches for resolution. To address these challenges, we leverage the hyperbolic discrete Morse flow, a powerful framework that allows us to adapt traditional techniques to the unique characteristics of our problem. This framework is particularly suited for handling the intricacies of non-local terms and offers a structured way to examine the interplay between these terms and the volume constraint. Our approach includes a rigorous mathematical analysis combined with the development of novel techniques specifically designed to manage the complex interactions within the system. Through this comprehensive methodology, we aim to establish the necessary conditions under which weak solutions can exist for the given hyperbolic Kirchhoff-type problem. The findings from this study provide new insights into the behavior of such systems, potentially contributing to advancements in the broader field of partial differential equations and applied mathematics.