Global Existence Of Weak Solutions Research Articles

Existence and exponential stability of solutions for a Balakrishnan–Taylor quasilinear wave equation with strong damping and localized nonlinear damping

Abstract In the paper, we study a Balakrishnan–Taylor quasilinear wave equation | z t | α ⁢ z t ⁢ t - Δ ⁢ z t ⁢ t - ( ξ 1 + ξ 2 ⁢ ∥ ∇ ⁡ z ∥ 2 + σ ⁢ ( ∇ ⁡ z , ∇ ⁡ z t ) ) ⁢ Δ ⁢ z - Δ ⁢ z t + β ⁢ ( x ) ⁢ f ⁢ ( z t ) + g ⁢ ( z ) = 0 |z_{t}|^{\alpha}z_{tt}-\Delta z_{tt}-\bigl{(}\xi_{1}+\xi_{2}\|\nabla z\|^{2}+% \sigma(\nabla z,\nabla z_{t})\bigr{)}\Delta z-\Delta z_{t}+\beta(x)f(z_{t})+g(% z)=0 in a bounded domain of ℝ n {\mathbb{R}^{n}} with Dirichlet boundary conditions. By using Faedo–Galerkin method, we prove the existence of global weak solutions. By the help of the perturbed energy method, the exponential stability of solutions is also established.

Weak solutions for a degenerate phase‐field model via Galerkin approximation

This paper is concerned with a phase‐field model driven by interface diffusion, which is an elliptic and nonlinear degenerate parabolic coupled system arising in a theory of interface motion in the elastically deformable materials. Spinodal decomposition is a typical example for this motion. We shall study the global existence of weak solutions to this model with a general polynomial potential function via the Galerkin method. Moreover, we perform the numerical simulations to investigate the spinodal decomposition process of binary alloy by applying this model.

On blow-up and on global existence of weak solutions to Cauchy problem for some nonlinear equation of the pseudoparabolic type

It is a brief exposition of results of the investigation of Cauchy problem for some nonlinear equation of pseudoparabolic type that is a generalisation of some model of semiconductor theory. In the paper, the potential theory for the linear part of the equation is elaborated, which demanded quite intricate technique, which can be used in other equations. The properties of the fundamental solution of this linear part are also of interest, because of the singularity of its 1st time derivative. This is not usual for this type of equations. Also, we obtain sufficiant conditions of solvability and of finite-time blow-up.

Long time behavior of weak solutions for the nonlinear damped beam equation

In this paper, we consider a nonlinear damped beam equation with logarithmic source and memory term. By using Galerkin method, logarithmic Sobolev inequality, and some compactness arguments, the global existence of weak solutions is obtained. Furthermore, we show the decay estimate of related energy under the smallness assumption on the initial data.

On the Existence of Global Compactly Supported Weak Solutions of the Vlasov–Poisson System with an External Magnetic Field

We consider the first mixed problem for the Vlasov–Poisson system with an external magnetic field in a domain with piecewise smooth boundary. This problem describes the kinetics of a two-component high-temperature plasma under the influence of a self-consistent electric field and an external magnetic field. The existence of global weak solutions is proved. In the case of a cylindrical domain, sufficient conditions are obtained for the existence of global weak solutions with supports in a strictly internal cylinder; this corresponds to the confinement of high-temperature plasma in a mirror trap.

On Blow-up and Global Existence of Weak Solutions to Cauchy Problem for Some Nonlinear Equation of the Pseudoparabolic Type

On Blow-up and Global Existence of Weak Solutions to Cauchy Problem for Some Nonlinear Equation of the Pseudoparabolic Type

Global Existence of Weak Solutions to 3D Compressible Primitive Equations of Atmospheric Dynamics with Degenerate Viscosity

In this work, we show the existence of global weak solutions to the three-dimensional compressible primitive equations of atmospheric dynamics with degenerate viscosity density-dependent for large initial data. With a pressure law of the form ρ2, we represent the vertical velocity as a function of the density and the horizontal one which will be important in using Faedo-Galerkin method to obtain the global existence of the approximate solutions. In analogy with the cases in [28] and [26, 27], we prove that the weak solutions satisfy the basic energy inequality and the Bresch-Desjardins entropy inequality. Based on these estimates and using compactness arguments, we prove the global existence of weak solutions of (1.1) by vanishing the parameters in our approximate system step by step.

On global weak solutions of the Vlasov-Poisson equations with external magnetic field

We consider the first mixed problem for the system of Vlasov-Poisson equations with a given external magnetic field in a bounded domain. This problem describes the kinetics of high-temperature plasma in controlled thermonuclear fusion plants and is considered with respect to unknown functions: electric field potential, distribution functions of positively charged ions and electrons. Additionally, we assumed that the distribution functions of charged particles satisfy the condition of mirror reflection from the boundary of the domain under consideration. We prove the existence of global weak solutions of such a problem.

Global existence of weak solutions to a BGK model relaxing to the barotropic Euler equations

We establish the global-in-time existence of weak solutions to a variant of the BGK model proposed by Bouchut (1999) which leads to the barotropic Euler equations, where the pressure is given by p(ρ)=ργ with γ∈(1,1+2/(d+2)], in the hydrodynamic limit. For the existence of solutions, we deal with γ∈(1,3] for the one-dimensional case, and γ∈(1,1+2/(d+2)]∪{1+2/d} for the multi-dimensional case. In particular, our existence theory makes the quantified estimates of hydrodynamic limit from the BGK-type equations to the multi-dimensional barotropic Euler system discussed by Berthelin and Vasseur (2005) completely rigorous.

Global existence and blow-up in higher-dimensional Patlak-Keller-Segel system for multi populations

This work is concerned with a degenerate parabolic-elliptic Patlak-Keller-Segel system for multi populations in space dimensions d≥3. We first provide a sufficient condition for the global existence of weak solution to the Cauchy problem. Then the global results are obtained both for any large initial data in the sub-critical case and for small initial data in the super-critical case. Finally, the finite-time blow-up solutions are constructed for large initial data in the super-critical case.

Global existence and uniqueness of weak solutions of a Stokes-Magneto system with fractional diffusions

We consider a Stokes-Magneto system in Rd (d≥2) with fractional diffusions Λ2αu and Λ2βb for the velocity u and the magnetic field b, respectively. Here α,β are positive constants and Λs=(−Δ)s/2 is the fractional Laplacian of order s. We establish global existence of weak solutions of the Stokes-Magneto system for any initial data in L2 when α, β satisfy 1/2<α<(d+1)/2, β>0, and min⁡{α+β,2α+β−1}>d/2. It is also shown that weak solutions are unique if β≥1 and min⁡{α+β,2α+β−1}≥d/2+1, in addition.

Global regularity and asymptotic stabilization for the incompressible Navier–Stokes-Cahn–Hilliard model with unmatched densities

We study an initial-boundary value problem for the incompressible Navier–Stokes–Cahn–Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\rightarrow \\infty $$\\end{document}. More precisely, the concentration function ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document} is a strong solution of the Cahn–Hilliard equation for (arbitrary) positive times, whereas the velocity field u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{{u}}}$$\\end{document} becomes a strong solution of the momentum equation for large times. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn–Hilliard equation with divergence-free velocity belonging only to L2(0,∞;H0,σ1(Ω))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(0,\\infty ; {\ extbf{H}}^1_{0,\\sigma }(\\Omega ))$$\\end{document}, the energy dissipation of the system, the separation property for large times, a weak-strong uniqueness type result, and the Lojasiewicz–Simon inequality. Additionally, in two dimensions, we show the existence and uniqueness of global strong solutions for the full system. Finally, we discuss the existence of global weak solutions for the case of the double obstacle potential.

Existence and time asymptotic profiles of weak solutions to the nonhomogeneous magneto‐micropolar equations

This paper studies the existence and decay estimates of weak solutions to the 3D nonhomogeneous magneto‐micropolar equations. First, the global existence of weak solutions are obtained. Then, we derive the and decay rates of weak solutions . Finally, the decay rates of magnetic field is obtained.

A generalized Duffing equation with the Coulomb’s friction law and Signorini–type contact conditions

This work provides mathematical and numerical analyses for a spring–mass system, in which Signorini–type contact conditions and Coulomb’s friction law with thermal effects are taken into consideration. The motion of a mass attached to a viscoelastic (Kelvin–Voigt type) nonlinear spring is described by a generalized Duffing equation. Signorini contact conditions are understood as extended complementarity conditions (CCs), where convolution is incorporated, allowing to consider thermal aspects of an obstacle. We prove the existence of global weak solutions for the highly nonlinear differential equation system with all the conditions, based on the regularized differential equation and the normal compliance condition with the standard mollifier. In addition, we investigate what side effects produce higher singularities of contact forces in dynamic contact problems, which is also supported by numerical evidences. Numerical schemes are proposed and then several groups of data are selected for the display of our numerical simulations.

Dynamics of a spatially homogeneous Vicsek model for oriented particles on a plane

We consider a spatially homogeneous Kolmogorov–Vicsek model in two dimensions, which describes the alignment dynamics of self-driven stochastic particles that move on a plane at a constant speed, under space-homogeneity. In [A. Figalli, M.-J. Kang and J. Morales, Global well-posedness of spatially homogeneous Kolmogorov–Vicsek model as a gradient flow, Arch. Rational Mech. Anal. 227 (2018) 869–896] Alessio Figalli and the authors have shown the existence of global weak solutions for this two-dimensional model. However, no time-asymptotic behavior is obtained for the two-dimensional case, due to the failure of the celebrated Bakery and Emery condition for the logarithmic Sobolev inequality. We prove exponential convergence (with quantitative rate) of the weak solutions towards a Fisher-von Mises distribution, using a new condition for the logarithmic Sobolev inequality.

Three-species drift-diffusion models for memristors

A system of drift-diffusion equations for the electron, hole, and oxygen vacancy densities in a semiconductor, coupled to the Poisson equation for the electric potential, is analyzed in a bounded domain with mixed Dirichlet–Neumann boundary conditions. This system describes the dynamics of charge carriers in a memristor device. Memristors can be seen as nonlinear resistors with memory, mimicking the conductance response of biological synapses. In the fast-relaxation limit, the system reduces to a drift-diffusion system for the oxygen vacancy density and electric potential, which is often used in neuromorphic applications. The following results are proved: the global existence of weak solutions to the full system in any space dimension; the uniform-in-time boundedness of the solutions to the full system and the fast-relaxation limit in two space dimensions; the global existence and weak–strong uniqueness analysis of the reduced system. Numerical experiments in one space dimension illustrate the behavior of the solutions and reproduce hysteresis effects in the current–voltage characteristics.

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.

Large time behavior of weak solutions to the surface growth equation

This paper studies the existence and decay estimates of weak solutions to the surface growth equation. First, the global existence of weak solutions is obtained by the approximation method introduced by Majda and Bertozzi [Vorticity and Incompressible Flow (Cambridge University Press, 2001)]. Then, we derive the L2-decay rates of weak solutions via the Fourier splitting method under the assumption that u0∈L1(R)∩L2(R). For more general cases, i.e., u0∈L2(R), the behavior of weak solutions in L2 is obtained by the spectral theory of self-adjoint operators.

Existence of weak solutions to a diffuse interface model involving magnetic fluids with unmatched densities

In this article we prove the global existence of weak solutions for a diffuse interface model in a bounded domain (both in 2D and 3D) involving incompressible magnetic fluids with unmatched densities. The model couples the incompressible Navier–Stokes equations, gradient flow of the magnetization vector and the Cahn–Hilliard dynamics describing the partial mixing of two fluids. The density of the mixture depends on an order parameter and the modelling (specifically the density dependence) is inspired from Abels et al. (Models Methods Appl Sci 22(3):1150013, 2011).

Propagation of Rough Initial Data for Navier–Stokes Equation

The purpose of this paper is to study the quantitative structure of a weak solution for an initial value problem of the compressible Navier–Stokes equation in the class of BV functions. In [T.-P. Liu and S.-H. Yu, Comm. Pure Appl. Math., 75 (2022), pp. 223–348] the precise pointwise structures of Green’s functions for various linearized problems are established, which serve as basic tools to establish the global existence of weak solutions to the Navier–Stokes equations with initial data as a small perturbation of a constant state in the BV class. In this paper, we obtain the time asymptotic pointwise structure of the weak solutions. It thus completes the initial motivation for the earlier paper on viewing the solution behavior as part of the well-posedness theory.