Uniqueness Of Weak Solutions Research Articles

Unique local weak solutions of the non-resistive MHD equations in homogeneous Besov space

ABSTRACT In this paper, the local existence and uniqueness of weak solutions to a d-dimensional non-resistive MHD equations in homogeneous Besov spaces are studied. Specifically we obtain the local existence of a weak solution ( u , b ) of the non-resistive MHD equations for the initial data u 0 ∈ B ˙ p , 1 d p − 1 ( R d ) and b 0 ∈ B ˙ p , 1 d p ( R d ) with 1 ≤ p ≤ ∞ , and the uniqueness of the weak solution when 1 ≤ p ≤ 2 d . Compared with the previous results for the non-resistive MHD equations, in the local existence part, the range of p extends to 1 ≤ p ≤ ∞ from 1 ≤ p ≤ 2 d , but the uniqueness of the solution requires 1 ≤ p ≤ 2 d yet.

Remarks on the linear wave equation

We make some remarks on the linear wave equation concerning the existence and uniqueness of weak solutions, satisfaction of the energy equation, growth properties of solutions, the passage from bounded to unbounded domains, and reconciliation of different representations of solutions.

Long-time convergence of a nonlocal Burgers’ equation towards the local N-wave

We study the long-time behaviour of the unique weak solution of a nonlocal regularisation of the (inviscid) Burgers equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic profile is given by the ‘N-wave’ entropy solution of the Burgers equation. The key ingredients of the proof are a suitable scaling argument and a nonlocal Oleinik-type estimate.

Insight into the Eyring–Powell fluid flow model using degenerate operator: geometric perturbation

This work provides a formulation of a fluid flow under a nonlinear diffusion based on a viscosity of Eyring–Powell type along with a degenerate semi-parabolic operator. The introduction of such a degenerate operator is significant as it allows us to explore a further general model not previously considered in the literature. Our aims are hence to provide analytical insights and numerical assessments to the mentioned flow model: firstly, some results are provided in connection with the regularity and uniqueness of weak solutions. The problem is converted into the travelling wave domain where solutions are obtained within an asymptotic expansion supported by the geometric perturbation theory. Finally, a numerical process is considered as the basis to ensure the validity of the analytical assessments presented. Such numerical process is performed for low Reynolds numbers given in classical porous media. As a main finding to highlight: we show that there exist exponential profiles of solutions for the velocity component. This result is not trivial for the non-linear viscosity terms considered.

New general decay results for a fully dynamic piezoelectric beam system with fractional delays under memory boundary controls

In this paper, we investigate the general stability results of a fully dynamic piezoelectric beam system with internal fractional delays under memory boundary controls. Firstly, by introducing two new equations and using two Volterra's inverse operators to deal with fractional delay terms and boundary memory terms, respectively, a new equivalent system is obtained. Secondly, by combining Faedo–Galerkin approximations with the compactness argument, we prove the local existence and uniqueness of weak solution. Finally, under a broader class of kernel functions, by constructing Lyapunov functionals and new equations and applying some technical lemmas, we establish the optimal and explicit energy decay results for two cases, namely, initial values and without imposing initial values equal to 0, that is, . It is worth pointing out that we focus more on the analysis of , because the stability results of are special cases of . This is the first time to analyze the stability of strong coupling problem by combining internal fractional delay and boundary viscoelastic damping. The obtained general decay results are not necessarily exponential or polynomial, which generalize the relevant results in the existing literature.

Global solutions and blow-up for Klein–Gordon equation with damping and logarithmic terms

In this paper, the initial boundary value problem for Klein–Gordon equation with weak and strong damping terms and nonlinear logarithmic term is investigated, which is known as one of the nonlinear wave equations in relativistic quantum mechanics and quantum field theory. Firstly, we prove the local existence and uniqueness of weak solution by using the Galerkin method and Contraction mapping principle. The global existence, energy decay and finite time blow-up of the solution with subcritical initial energy are established. Then these conclusions are extended to the critical initial energy. Besides, the finite time blow-up result with supercritical initial energy is shown.

Regularity and solutions for flame modelling in porous medium

The presented article deals with a model of flame propagation in porous medium. We depart from previously reported models in flame propagation, and we propose a new modelling conception based on a p-Laplacian operator. Such an operator is intended to extend the conceptions for reproducing the diffusion given in porous media. In addition, we introduce a nonlinear reaction term that generalizes the classical KPP-Fisher model of typical use in combustion. Our analysis shows first the boundedness and uniqueness of weak solutions. Afterward, we reformulate the driving equations using the travelling wave technique; and we introduce a density and flux ansatz to analyse the stability of a critical point. We obtain asymptotic profiles of travelling wave solutions, by making use of the Geometric Perturbation Theory. Eventually, we obtain upper solutions under selfsimilar forms for finite support propagating flames. This is particularly applicable for flames departing from compactly supported initial pressure–temperature distributions.

Global existence of unique weak solutions and decay rates of Active model B with the logarithmic Cahn–Hilliard equation in Wiener space

In this paper, we deal with Active model B with the logarithmic Cahn–Hilliard equation. Under some smallness assumptions to initial data, we prove that there is a unique global-in-time solution which decays exponentially in time.

Schrödinger equations defined by a class of self-similar measures

We study linear and non-linear Schrödinger equations defined by fractal measures. Under the assumption that the Laplacian has compact resolvent, we prove that there exists a unique weak solution for a linear Schrödinger equation, and then use it to obtain the existence and uniqueness of a weak solution of a non-linear Schrödinger equation. We prove that for a class of self-similar measures on \mathbb{R} with overlaps, the linear Schrödinger equations can be discretized so that the finite element method can be applied to obtain approximate solutions. We also prove that the numerical solutions converge to the actual solution and obtain the rate of convergence.

Asymptotically almost periodic synchronization in fuzzy competitive neural networks with Caputo-Fabrizio operator

This paper researches a class of Caputo-Fabrizio fractional-order fuzzy competitive neural networks (CF-FCNNs) with different time scales and time delays. Firstly, according to Banach contracting mapping principle and some appropriate assumptions, it derives CF-FCNNs possess a unique weak solution. Then, exponential Euler difference scheme of discrete-time CF-FCNNs (DCF-FCNNs) is presented by the constant variation methods and the difference format is implicit Euler difference and is solved by fractional PECE algorithm. Next, the existence of a unique bounded asymptotically almost periodic sequence solution, global exponential stability and synchronization of DCF-FCNNs are considered. In the end, numerical examples are given to describe the effectiveness of the acquired conclusions.

A triangle-based positive semi-discrete Lagrangian–Eulerian scheme via the weak asymptotic method for scalar equations and systems of hyperbolic conservation laws

In this paper, we show how to construct a positive semi-discrete Lagrangian–Eulerian scheme on triangular grids that asymptotically satisfies multidimensional scalar hyperbolic problems and first-order nonlinear systems of conservation laws. The construction of such numerical scheme is based on the improved concept of space–time no-flow curves. For scalar equations, through a sequence of computational grid functions that tend to satisfy the underlying hyperbolic conservation law(s) in a weak sense in the space variable and in a strong sense in the time variable, we prove that the proposed scheme converges to the unique weak solution satisfying a type of Kruzhkov entropy condition. For nonlinear systems of first-order hyperbolic conservation laws, we show that the proposed scheme also satisfies a (weak) positivity principle. This provides an effective numerical analysis tool (based on weak asymptotic methods) to construct a family of continuous functions that asymptotically satisfies scalar conservation laws with discontinuous nonlinearity and systems that have irregular solutions. In order to assess the robustness of the proposed approach, we solve a series of numerical tests on several hyperbolic problems (scalar and systems) using the positive semi-discrete Lagrangian–Eulerian scheme on triangular grids. The results confirm that the proposed method is efficient (since it is Riemann-solver-free) and provides good resolution for scalar models and for the solution of systems of conservation laws.

The uniqueness solution for a fractional φ-Laplacian Dirichlet problem and its spectrum

In this paper, we demonstrate the existence of a unique weak solution for a nonlocal Kirchhoff problem under Dirichlet boundary conditions, involving the fractional φ-Laplacian operator. Our major outcome is acquired by applying variational approaches and critical points theory. In addition, we analyse the spectrum and the eigenvalues associated to this problem. At the end and under some assumptions, we give an application to the previous problems.

Two-Phase Flows with Bulk–Surface Interaction: Thermodynamically Consistent Navier–Stokes–Cahn–Hilliard Models with Dynamic Boundary Conditions

We derive a novel thermodynamically consistent Navier–Stokes–Cahn–Hilliard system with dynamic boundary conditions. This model describes the motion of viscous incompressible binary fluids with different densities. In contrast to previous models in the literature, our new model allows for surface diffusion, a variable contact angle between the diffuse interface and the boundary, and mass transfer between bulk and surface. In particular, this transfer of material is subject to a mass conservation law including both a bulk and a surface contribution. The derivation is carried out by means of local energy dissipation laws and the Lagrange multiplier approach. Next, in the case of fluids with matched densities, we show the existence of global weak solutions in two and three dimensions as well as the uniqueness of weak solutions in two dimensions.

Strong convergence of weighted gradients in parabolic equations and applications to global generalized solvability of cross-diffusive systems

In the first part of the present paper, we show that strong convergence of (v_{0 varepsilon })_{varepsilon in (0, 1)} in L^1(Omega ) and weak convergence of (f_{varepsilon })_{varepsilon in (0, 1)} in L_{text {loc}}^1({{overline{Omega }}} times [0, infty )) not only suffice to conclude that solutions to the initial boundary value problem vεt=Δvε+fε(x,t)inΩ×(0,∞),∂νvε=0on∂Ω×(0,∞),vε(·,0)=v0εinΩ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} v_{\\varepsilon t} = \\Delta v_\\varepsilon + f_\\varepsilon (x, t) &{} \ ext {in }\\Omega \ imes (0, \\infty ), \\\\ \\partial _\ u v_\\varepsilon = 0 &{} \ ext {on }\\partial \\Omega \ imes (0, \\infty ), \\\\ v_\\varepsilon (\\cdot , 0) = v_{0 \\varepsilon } &{} \ ext {in }\\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}which we consider in smooth, bounded domains Omega , converge to the unique weak solution of the limit problem, but that also certain weighted gradients of v_varepsilon converge strongly in L_{text {loc}}^2({{overline{Omega }}} times [0, infty )) along a subsequence. We then make use of these findings to obtain global generalized solutions to various cross-diffusive systems. Inter alia, we establish global generalized solvability of the system ut=Δu-χ∇·(uv∇v)+g(u),vt=Δv-uv,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\Delta u - \\chi \ abla \\cdot (\\frac{u}{v} \ abla v) + g(u), \\\\ v_t = \\Delta v - uv, \\end{array}\\right. } \\end{aligned}$$\\end{document}where chi > 0 and g in C^1([0, infty )) are given, merely provided that (g(0) ge 0 and) -g grows superlinearly. This result holds in all space dimensions and does neither require any symmetry assumptions nor the smallness of certain parameters. Thereby, we expand on a corresponding result for quadratically growing -g proved by Lankeit and Lankeit (Nonlinearity 32(5):1569–1596, 2019).

Existence and uniqueness of a weak solution to fractional single-phase-lag heat equation

In this article, we study the existence and uniqueness of a weak solution to the fractional single-phase lag heat equation. This model contains the terms $$\mathcal {D}_t^\alpha (\partial _{t}u)$$ and $$\mathcal {D}_t^\alpha u $$ (with $$\alpha \in (0,1)$$ ), where $$\mathcal {D}_t^\alpha $$ denotes the Caputo fractional differential operator (in time) of order $$\alpha $$ . We consider homogeneous Dirichlet boundary data for the temperature. We rigorously show the existence of a unique weak solution under low regularity assumptions on the data. Our main strategy is to use the variational formulation and a semidiscretisation in time based on Rothe’s method. We obtain a priori estimates on the discrete solutions and show convergence of the Rothe functions to a weak solution. The variational approach is employed to show the uniqueness of this weak solution to the problem. We also consider the one-dimensional problem and derive a representation formula for the solution. We establish bounds on this explicit solution and its time derivative by extending properties of the multivariate Mittag-Leffler function.

High-order iterative scheme for Kirchhoff-type wave equation with the source containing three unknown values

In this paper, a high-order iterative scheme is established in order to get a convergent sequence at a rate of order N, (N ≥ 2) to a local unique weak solution of a nonlinear Kirchhoff-type wave equation associated with Robin conditions.

Global solution and blow-up for a class of Δγ-biharmonic parabolic equations with logarithmic nonlinearity

In this paper, we study a class of [Formula: see text]-biharmonic parabolic equations with logarithmic nonlinearity. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique and the Sobolev inequality, we obtain the existence of the unique global weak solutions. In addition, we also provide sufficient conditions for the large time decay of global weak solutions and the finite time blow-up of weak solutions.

Analysis and Numerical Approach of a Coupled Thermo-Electro-Mechanical System for Nonlinear Hencky-Type Materials with Nonlocal Coulomb’s Friction

The effects of an included temperature field in the contact process between a piezoelectric body and a rigid foundation with thermal and electrical conductivity are discussed. The constitutive relation of the material is assumed to be thermo-electro-elastic and involves the nonlinear elastic constitutive Hencky’s law. The frictional contact is modeled with Signorini’s conditions, the regularized Coulomb law, and the regularized electrical and thermal conductivity conditions. The resulting thermo-electromechanical model includes the temperature field as an additional state variable to take into account thermal effects alongside with those of the piezoelectric. The existence of the unique weak solution to the problem is established by using Schauder’s fixed point theorem combined with arguments from the theory of variational inequalities involving nonlinear strongly monotone Lipschitz continuous operators. A successive iteration technique to solve the problem numerically is proposed, and its convergence is established. A variant of the Augmented Lagrangian, the so-called Alternating Direction Method of multipliers (ADMM), is used to decompose the original problem into two sub-problems, solve them sequentially and update the dual variables at each iteration. The influence of the thermal boundary condition on the behavior of contact forces and electrical potential is shown through graphical illustrations.

Global existence for the 2D anisotropic Bénard equations with partial variable viscosity

The current paper aims to study the two‐dimensional (2D) Bénard equations with variable‐viscosity, where we obtain the global existence of a unique weak solution to this system without any assumptions of small initial data.

The higher order nonlinear Schrödinger equation with quadratic nonlinearity on the real axis

The initial value problem is considered for a higher order nonlinear Schrödinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional damping large-time decay of solutions without any smallness assumptions is also established. The main difficulty of the study is the non-smooth character of the nonlinearity.