Global Weak Solutions Research Articles

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.

Non-existence of global weak solutions to semi-linear wave equations involving time-dependent structural damping terms

Non-existence of global weak solutions to semi-linear wave equations involving time-dependent structural damping terms

Global weak solutions to the compressible magnetohydrodynamic and Vlasov equations

Global weak solutions to the compressible magnetohydrodynamic and Vlasov equations

Almost sure existence of global weak solutions for supercritical electron MHD

Almost sure existence of global weak solutions for supercritical electron MHD

Uniqueness of global weak solutions to the frame hydrodynamics for biaxial nematic phases in $\mathbb{R}^2$

Uniqueness of global weak solutions to the frame hydrodynamics for biaxial nematic phases in $\mathbb{R}^2$

Eventual smoothness and stabilization of global weak solutions to a chemotaxis system with sublinear consumption of chemoattractant

Eventual smoothness and stabilization of global weak solutions to a chemotaxis system with sublinear consumption of chemoattractant

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

<abstract><p>In spaces with any dimension, a class of logarithmic wave equations with Balakrishnan-Taylor damping, nonlinear weak damping and strong linear damping was considered</p> <p><disp-formula> <label/> <tex-math id="FE1"> \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} </tex-math></disp-formula></p> <p>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.</p></abstract>

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

<p>This paper deals with the initial boundary value problem for a class of <italic>n</italic>-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.</p>

The Absence of Global Weak Solutions for a Quasilinear Parabolic Differential Inequality in Exterior Domain

The Absence of Global Weak Solutions for a Quasilinear Parabolic Differential Inequality in Exterior Domain

Existence of large-data global weak solutions to kinetic models of nonhomogeneous dilute polymeric fluids

We prove the existence of large-data global-in-time weak solutions to a general class of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for nonhomogeneous incompressible dilute polymeric fluids in a bounded domain in $ \mathbb{R}^d $, $ d = 2 $ or $ 3 $. The class of models under consideration involves the Navier–Stokes system with variable density, where the viscosity coefficient depends on both the density and the polymer number density, coupled to a Fokker–Planck equation with a density-dependent drag coefficient. The proof is based on combining a truncation of the probability density function with a two-stage Galerkin approximation and weak compactness and compensated compactness techniques to pass to the limits in the sequence of Galerkin approximations and in the truncation level.

Global weak solutions for a nonlocal multispecies Fokker–Planck–Landau system

Global weak solutions for a nonlocal multispecies Fokker–Planck–Landau system

Global weak solutions to a 3D/3D fluid-structure interaction problem including possible contacts

In this paper, we study an interaction problem between a 3D compressible viscous fluid and a 3D nonlinear viscoelastic solid fully immersed in the fluid, coupled together on the interface surface. The solid is allowed to have self-contact or contact with the rigid boundary of the fluid container. For this problem, a global weak solution with defect measure is constructed by using a multi-layered approximation scheme which decouples the body and the fluid by penalizing the fluid velocity and allowing the fluid to pass through the body, while the body is supplemented with a contact-penalization term. The resulting defect measure is a consequence of pressure concentrations that can appear where the fluid meets the (generally irregular) points of self-contact of the solid. Moreover, we study some geometrical properties of the fluid-structure interface and the contact surface. In particular, we prove a lower bound on area of the interface.

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.

Global weak solutions in nonlinear 3D thermoelasticity

Global weak solutions in nonlinear 3D thermoelasticity

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.

Derivation of a 2D PCCU-AENO method for nonconservative problems

In this paper, we introduce a methodology to design genuinely two-dimensional (2D) second-order well-balanced path-conservative central-upwind (PCCU) schemes. The scheme studies dam-break with high sediment concentration over abrupt moving topography quickly spatially variable even in the presence of the resonance. This study is possible via a 2D hyperbolic sediment transport model (including arbitrarily sloping sediment beds and associated energy and entropy) in new generalized Shallow Water equations derived with associated energy and entropy in this work. We expose some properties of the model and we establish an existence result of global weak solutions. In addition, we show the convergence of a sequence of solutions of the proposed new model. The second-order accuracy of the PCCU scheme is achieved using a new extension AENO (Averaging Essentially Non-Oscillatory) reconstruction developed in the 2D version of this work. We prove that the derived 2D scheme is well-balanced, positivity-preserving and steady states capturing. Several dam break tests are made to show the ability and the superb performances of the proposed numerical modeling. The proposed method can help to address many engineering problems.

Global weak solutions of an initial-boundary value problem on a half-line for the higher order nonlinear Schrödinger equation

An initial-boundary value problem with one boundary condition is considered for the higher order nonlinear Schrödinger equation. It is assumed that either the boundary condition is homogeneous or the nonlinearity in the equation is quadratic. Results on existence, uniqueness and continuous dependence on input data of global weak solutions are obtained.

Superlinear degradation in a doubly degenerate nutrient taxis system

This work is concerned with the doubly degenerate cross-diffusion system with generalized logistic source ut=∇⋅(uv∇u)−∇⋅(u2v∇v)+f(u),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0in a smoothly bounded domain Ω⊂Rn(n≥2) with no-flux boundary conditions, where f(u)=ρu−μuκ with ρ,μ>0. It is shown that under the assumption that u0∈W1,∞(Ω) is nonnegative with u0⁄≡0 and v0∈W1,∞(Ω) is positive in Ω̄, if κ>(n+2)/2,then the problem admits a global weak solution.

Output tracking for nonlinear stochastic systems driven by bounded and colored noises

In this paper, we consider practical tracking problem for a class of nonlinear stochastic systems driven by bounded and colored noises by means of active disturbance rejection control approach. The nonlinear extended state observer with homogeneous weighted functions is constructed to estimate in real time the unmeasurable state and the total stochastic disturbance that affects system performance. An observer-based output feedback is designed to guarantee the almost surely practical convergence of the tracking errors. The existence of the global weak solution of the closed-loop system is proved and the tracking convergence accuracy is shown to be higher than that of linear one under the same tuning parameters. Some numerical simulations are carried out to validate the effectiveness of the proposed controller.

Existence and qualitative analysis of a fully cross-diffusive predator-prey system with nonlinear taxis sensitivity

This paper investigates a fully cross-diffusive predator-prey system with nonlinear taxis sensitivity(⋆){ut=D1uxx−χ1(uk1vx)x+μ1u(λ1−u+a1v),vt=D2vxx+χ2(vk2ux)x+μ2v(λ2−v−a2u), under homogeneous boundary conditions of Neumann type for u and v, in an open bounded interval Ω⊂R, where Di,χi,λi,ai>0 and μi≥0 for i∈{1,2}. In Tao and Winkler (2021) [1]; (2022) [2], Tao and Winkler studied the system (⋆) with k1=k2=1 and obtained the global existence and asymptotic behavior. We study the system (⋆) with k1,k2≠1 and prove that:•If k1∈(1,4031) and k2∈[14,12−9k18−5k1), then the model (⋆) possesses global weak solutions for arbitrarily large positive initial data in H1(Ω).•If k1∈(1,1110) and k2∈[23,12−9k18−5k1), then the global weak solution (u,v) of (⋆) with μ1=μ2=0 stabilizes toward homogeneous equilibria (1|Ω|∫Ωu0,1|Ω|∫Ωv0).•If k1∈(1,98) and k2∈[12,12−9k18−5k1), then the global weak solution (u,v) of (⋆) with μ1,μ2>0 converges to constant stable steady state (λ1,0) provided that λ2≤a2λ1 and the tactic sensitivities χ1 and χ2 are suitably small.