Global Existence Of Weak Solutions Research Articles

Global existence of weak solutions to the two‐dimensional nematic liquid crystal flow with partially free boundary

AbstractWe consider a nematic liquid crystal flow with partially free boundary in a smooth bounded domain in . We prove regularity estimates and the global existence of weak solutions enjoying partial regularity properties, and a uniqueness result.

Existence of global weak solution to tumor chemotaxis competition systems with loop and signal dependent sensitivity

This article examines the weak solution of a fully parabolic chemotaxis-competition system with loop and signal-dependent sensitivity. The system is subject to homogeneous Neumann boundary conditions within an open, bounded domain \(\Omega\subset\mathbb{R}^n\), where \(n\geq 1\) and \(\partial\Omega\) is smooth. We assume that the parameters in the system are positive constants. Additionally, the initial data \((u_{10}, u_{20}, v_{10}, v_{20})\in L^2(\Omega)\times L^2(\Omega) \times W^{1,2}(\Omega)\times W^{1,2}(\Omega)\) are non-negative. The existence of a weak solution to the problem is established using energy inequality method. For more information see https://ejde.math.txstate.edu/Volumes/2024/56/abstr.html

Existence of global weak solutions and simulations to a Dirichlet problem for a generalized Swift–Hohenberg equation

In this paper, we shall investigate an initial–boundary value problem of a generalized Swift–Hohenberg model subject to homogeneous Dirichlet boundary conditions in two spatial dimensions. The model consists of a nonlinear term of the form ψ2Δ2ψ2 in the free energy functional, which is used to model the stability of fronts between hexagons and squares in pinning effect. We first prove the global-in-time existence and uniqueness of weak solutions to this initial–boundary value problem in the case with the parameter β<0, where we employ the energy method and make use of various techniques to derive delicate a priori estimates. At the end, a few numerical experiments of the model are also performed to study the competition between hexagons and squares.

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.

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&amp;lt;p&amp;lt;∞,1≤q≤∞,1+n+2p&amp;lt;α&amp;lt;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.

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.

Non-Newtonian incompressible fluids with nonlinear shear tensor and hereditary conditions

We consider a mathematical model with delay for non-Newtonian incompressible fluids in a bounded domain. Existence of global weak solutions is proved under suitable regularity on the initial data and the forces. Conditions for uniqueness are also given, but in general the results are stated in a multi-valued framework. Suitable multi-valued dynamical systems are well-posed, using basically L2(Ω)n×L2(−h,0;L2(Ω)n) or C([−h,0];L2(Ω)n) norms. Then the existence of pullback attractors is ensured acting on several universes, some of them of fixed bounded sets and others of tempered type, depending on parameters related to an integrability condition of the force and the delay term. Finally, relationships between these families of attractors are also provided, improving the characterization of attraction with respect to previous results.

Formal derivation and existence of global weak solutions of an energetically consistent viscous sedimentation model

The purpose of this paper is to derive a viscous sedimentation model from the Navier-Stokes system for incompressible flows with a free moving boundary. The derivation is based on the different properties of the fluids; thus, we perform a multiscale analysis in space and time, and a different asymptotic analysis to derive a system coupling two different models: the sediment transport equation for the lower layer and the shallow water model for the upper one. We finally prove the existence of global weak solutions in time for model containing some additional terms.

Global existence of weak solutions and weak–strong uniqueness for nonisothermal Maxwell–Stefan systems *

The dynamics of multicomponent gas mixtures with vanishing barycentric velocity is described by Maxwell–Stefan equations with mass diffusion and heat conduction. The equations consist of the mass and energy balances, coupled to an algebraic system that relates the partial velocities and driving forces. The global existence of weak solutions to this system in a bounded domain with no-flux boundary conditions is proved by using the boundedness-by-entropy method. A priori estimates are obtained from the entropy inequality which originates from the consistent thermodynamic modelling. Furthermore, a conditional weak–strong uniqueness property is shown by using the relative entropy method.

On a nonlinear diffussive model for the evolution of cells within a moving domain

We investigate the dynamics of a nonlinear model describing the motion of cells under the effect of porous-medium diffusion and transport and in the presence of nutrient and drug application. The momentum equation for the evolution of the velocity field is governed by Darcy’s law, while the evolution of the chemical attractant (nutrient or drug) is governed by a diffusion equation. The system evolves within a moving domain in R 3 \mathbb {R}^3 accounting for the expansion or shrinkage of the tumor. The global existence of weak solutions is established with the aid of a regularized approximating scheme and an Arbitrary Lagrangian-Eulerian (ALE) mapping for the motion of the tumor. This work provides a variational framework suitable for both analysis and simulations.

On the Mass Transfer in the 3D Pitaevskii Model

We examine a micro-scale model of superfluidity derived by Pitaevskii (Sov. Phys. JETP 8:282-287, 1959) which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of global weak solutions in T3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}^3$$\\end{document} for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.

On the Existence of Global Weak Solutions to the 3D Electrically Conductive Rosensweig System and Their Convergence Towards Quasi-Equilibrium

In this article, we study an electrically conductive Rosensweig model for ferrofluids, whose Bloch–Torrey regularization was studied by Hamdache and Hamroun (Appl Math Optim 81(2):479–509, 2020). We mainly prove the global existence of weak solutions to the non-regularized model under a certain smallness condition on the electric conductivity. Hence, our result not only solves a problem that was left open by Hamdache and Hamroun, but it can also serve as a confirmation that ferrofluids are naturally poor conductors of electric current. The proof, which is interesting in itself, is quite involved and relies on the Helmohltz–Leray decomposition of the magnetic fields and the use of renormalized solutions for the magnetization. We also give a rigorous and detailed description of the convergence of the global weak solutions towards the quasi-equilibrium in the relaxation time limit regime τ→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au \\rightarrow 0$$\\end{document}.

A pseudo‐parabolic equation with logarithmic nonlinearity: Global existence and blowup of solutions

This article deals with a fourth‐order pseudo‐parabolic partial differential equation with logarithmic nonlinearity. We adopt the Faedo–Galerkin method to analyze the global existence of weak solutions and discuss the existence of a solution in subcritical and critical initial energy situations. Further, applying the concavity approach, we have shown that the solution blows up when the initial energy is subcritical, critical, and supercritical. Moreover, we have provided an upper and lower bound for blow‐up time and a decay estimate for the global solutions.

Optimal time decay of the compressible micropolar fluids with discontinuous initial data in 3D

The global existence of weak solutions to Cauchy problem for compressible micropolar fluids with discontinuous initial data has been established in [4]. However, the optimal decay rates of these weak solutions remain an open problem. The current work is to show the optimal decay rates of these weak solutions in Lr-norm with 2<r≤∞ and the optimal decay rates of the first order derivative of the velocity and micro-rotational velocity in L2-norm which are the same as the rates of classical solutions in [20]. In this process, we combine electromagnetic and fluid decomposition, frequency domain decomposition, and construct a series of temporal energy functional to solve complex nonlinear problems with low regularity.

Existence of global weak solutions of inhomogeneous incompressible Navier–Stokes system with mass diffusion

Existence of global weak solutions of inhomogeneous incompressible Navier–Stokes system with mass diffusion

On the Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Coupled Petrovsky-Wave System

In this paper, we consider the initial-boundary value problem for a class of nonlinear coupled wave equation and Petrovesky system in a bounded domain. The strong damping is nonlinear. First, we prove the existence of global weak solutions by using the energy method combined with Faedo-Galarkin method and the multiplier method. In addition, under suitable conditions on functions gi(⋅), i = 1, 2 and a(⋅), we obtain both exponential and polynomial decay estimates. The method of proofs is direct and based on the energy method combined with the multipliers technique, on some integral inequalities due to Haraux and Komornik.

Almost sure existence of global weak solutions for supercritical electron MHD

Almost sure existence of global weak solutions for supercritical electron MHD

Asymptotic behavior for a class of logarithmic wave equations with Balakrishnan-Taylor damping, nonlinear weak damping and strong linear damping

&lt;abstract&gt;&lt;p&gt;In spaces with any dimension, a class of logarithmic wave equations with Balakrishnan-Taylor damping, nonlinear weak damping and strong linear damping was considered&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ {u_{tt}} - M(t)\Delta u + \left( {g * \Delta u} \right)(t) + h({u_t}){u_t} - \Delta {u_t} + f(u) + u = u\ln {\left| u \right|^k} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;with Dirichlet boundary condition. Under a set of specified assumptions, we established the existence of global weak solutions and elucidated the decay rate of the energy function for particular initial data. This contribution extended and surpassed prior investigations, as documented by A. Peyravi (2020), which omitted the consideration of Balakrishnan-Taylor damping and strong linear damping while being confined to three spatial dimensions. Our findings underscored the pivotal role of these overlooked damping factors. Furthermore, our demonstration underscored that Balakrishnan-Taylor damping, weak damping and strong damping collectively induced an exponential decay, although the precise nature of this decay was contingent upon the differentiable function associated with the memory damping term $ {\zeta (t)} $. Consequently, the absence of the damping term in reference by A. Peyravi (2020) was unequivocally shown not to augment the decay rate. This insight enhanced our understanding of the nuanced dynamics involved and contributed to the refinement of existing models.&lt;/p&gt;&lt;/abstract&gt;

Global existence of weak solutions to a class of higher-order nonlinear evolution equations

&lt;p&gt;This paper deals with the initial boundary value problem for a class of &lt;italic&gt;n&lt;/italic&gt;-dimensional higher-order nonlinear evolution equations that come from the viscoelastic mechanics and have no positive definite energy. Through the analysis of functionals containing higher-order energy of motion, a modified potential well with positive depth is constructed. Then, using the potential well method, and Galerkin method, it has been shown that when the initial data starts from the stable set, there exists a global weak solution to such an evolution problem.&lt;/p&gt;

Global Existence of Weak Solutions for Quantum Magnetohydrodynamic Equations

Global Existence of Weak Solutions for Quantum Magnetohydrodynamic Equations