Global Weak Solutions Research Articles

Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities

In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we derive a growth estimate for the blowing up global solutions.

Global Weak Solutions to a Fluid-particle System of an Incompressible Non-Newtonian Fluid and the Vlasov Equation

Global Weak Solutions to a Fluid-particle System of an Incompressible Non-Newtonian Fluid and the Vlasov Equation

Global weak solutions with higher regularity to the compressible Navier-Stokes equations under the no-slip boundary conditions

Global weak solutions with higher regularity to the compressible Navier-Stokes equations under the no-slip boundary conditions

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.

Continuous solutions for a two-dimensional cross-diffusion problem involving doubly degenerate diffusion and logistic proliferation

Emerging from the modeling of intricate structures formed by bacterial motion in nutrient-deficient environments, the doubly degenerate nutrient taxis system [Formula: see text] is considered in smoothly bounded domains [Formula: see text]. In contrast to frameworks devoid of logistic growth where only small-data global solutions have so far been found in the literature, quadratic degradation in the logistic growth is shown to markedly amplify the overall dissipative effects to the extent that, for arbitrary smooth initial data, an associated no-flux type initial-boundary problem possesses global continuous weak solutions. Central to our findings is the identification of an advantageous dissipative structure subtly embedded in the system, the exploitation of which leads to the establishment of a positive lower bound for nutrients levels within any finite time interval, thereby ruling out the appearance of cross-degeneracies within such timeframes.

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 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.

A generalized Allen–Cahn model with mass source and its Cahn–Hilliard limit

AbstractThe present paper is concerned with a fourth‐order Allen–Cahn model with logarithmic potential and mass source that describes the process of phase separation in two‐component systems accompanied by a flux of material. The existence of a global weak solution is obtained under appropriate hypotheses on the source term. Furthermore, we study its Cahn–Hilliard limit as a small parameter goes to zero. The main difficulty in the mathematical analysis of the model lies in the presence of the source term that leads to the nonconservation of mass, contrary to the original Cahn–Hilliard theory.

Existence of global and explosive mild solutions of fractional reaction–diffusion system of semilinear SPDEs with fractional noise

In this paper, we investigate the existence and finite-time blow-up for the solution of a reaction–diffusion system of semilinear stochastic partial differential equations (SPDEs) subjected to a two-dimensional fractional Brownian motion given by [Formula: see text] for [Formula: see text], along with [Formula: see text] where [Formula: see text] is the fractional power [Formula: see text] of the Laplacian, [Formula: see text] and [Formula: see text] and [Formula: see text] are real constants. We provide sufficient conditions for the existence of a global weak solution. Under the assumption that [Formula: see text] with Hurst index [Formula: see text] we obtain the blow-up times for an associated system of random partial differential equations in terms of an integral representation of exponential functions of Brownian motions. Moreover, we provide lower and upper bounds for the finite-time blow-up of the above system of SPDEs and obtain the upper bounds for the probability of non-explosive solution to our considered system.

Global weak solutions to a time-periodic body-liquid interaction problem

We prove existence of time-periodic weak solutions to the coupled liquid-structure problem constituted by an incompressible Navier–Stokes fluid interacting with a rigid body of finite size, subject to an undamped linear restoring force. The fluid flow is generated by a uniform, time-periodic velocity field {\boldsymbol V} far from the body. We emphasize that our result is global, in the sense that no restriction is imposed on the magnitude of {\boldsymbol V} and, rather remarkably, the frequency of {\boldsymbol V} is entirely arbitrary. Thus, in particular, it can coincide with any multiple of a natural frequency of vibration of the body so that, with this model, resonance cannot occur. Although based on the classical “invading domains” technique, our approach requires several new ideas. Indeed, due to the lack of sufficient dissipation, it appears quite unfeasible to show the existence of a fixed point of the Poincaré map at the finite-dimensional level along the Galerkin approximant. Therefore, unlike the usual strategy, such a result must be proven directly in a class of weak solutions, and therefore in the infinite-dimensional framework.

Extinction and non-extinction of solutions for nonlocal fractional [formula omitted]-Kirchhoff problem with logarithmic nonlinearity

In this paper, we discuss the extinction and non-extinction properties of solutions for a class of Kirchhoff type parabolic problems involving fractional p-Laplacian and logarithmic nonlinearity. Based on energy estimates, embedding theorems, and certain ordinary differential inequalities, the global existence of solutions, and the extinction and non-extinction properties of global weak solutions are obtained. The results of this paper extend and complement previous research findings on this type of equation.

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We consider the stability of the global weak solution of the Quantum Euler system in two space dimensions. More precisely, we establish compactness properties of global finite energy weak solution for large initial data provided that these are axisymmetric. The main novelty is that the initial velocity is not necessary irrotational when the density is not vanishing, our main argument is based on the Madelung transform which enables us to prove new Kato estimates on the irrotational part of the velocity.

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 weak solutions to a nonlinear chemotaxis system with singular density-suppressed motility

In this work, we study the no-flux initial-boundary value problem for the migration-consumption taxis system involving singular density-suppressed motility [Formula: see text] in a bounded domain [Formula: see text] [Formula: see text], where [Formula: see text] generalizes the singular prototype given by [Formula: see text] [Formula: see text] with [Formula: see text]. We prove that if [Formula: see text] and [Formula: see text], then the model (⋆) possesses a global weak-strong solution.

Invariant Regions and Global Existence of Uniqueness Weak Solutions for Tridiagonal Reaction-Diffusion Systems

In this paper we study the existence of uniqueness global weak solutions for m × m reaction-diffusion systems for which two main properties hold: the positivity of the weak solutions and the total mass of the components are preserved with time. Moreover, we suppose that the non-linearities have critical growth with respect to the gradient. The technique we use here in order to prove global existence is in the same spirit of the method developed by Boccardo, Murat, and Puel for a single equation. Keywords: Semigroups, local weak solution, global weak solution, reaction- diffusion systems, invariant regions, matrice of diffusion.

Global weak solutions to a phase-field model for seawater solidification with melt convection

Global weak solutions to a phase-field model for seawater solidification with melt convection

Spatially-Periodic Solutions for Evolution Anisotropic Variable-Coefficient Navier–Stokes Equations: I. Weak Solution Existence

We consider evolution (non-stationary) spatially-periodic solutions to the n-dimensional non-linear Navier–Stokes equations of anisotropic fluids with the viscosity coefficient tensor variable in spatial coordinates and time and satisfying the relaxed ellipticity condition. Employing the Galerkin algorithm with the basis constituted by the eigenfunctions of the periodic Bessel-potential operator, we prove the existence of a global weak solution.

The horizontal magnetic primitive equations approximation of the anisotropic MHD equations in a thin 3D domain

In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter ε∈(0,∞) , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are μ=O(1) and ν=O(εα) , and the orders of magnetic diffusion coefficients κ and σ are κ=O(1) and σ=O(εα) , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For H1 -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For H1 -initial data with additional regularity (∂zA~0,∂zB~0)∈Lp(Ω)(2<p<∞) , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal. 272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For H2 -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order O(εγ/2) , where γ=min{2,α−2} with α∈(2,∞) . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order O(ε) .