Global Weak Solutions Research Articles

Weighted Hardy–Sobolev inequality and global existence result of thermoelastic system on manifolds with corner‐edge singularities

This article concerns with the thermoelastic corner‐edge‐type system with singular potential function on a wedge manifold with corner singularities. First, we introduce weighted p‐Sobolev spaces on manifolds with corner‐edge singularities. Then, we prove the corner‐edge‐type Sobolev inequality, Poincar inequality and Hardy inequality and obtain some results about the compactness of embedding maps on the weighted corner‐edge Sobolev spaces. Finally, as an application of these results, we apply the potential well theory and the Faedo–Galerkin approximations to obtain the global weak solutions for the thermoelastic corner‐edge‐type system 1.1.

Nonexistence of Global Solutions to Higher-Order Time-Fractional Evolution Inequalities with Subcritical Degeneracy

In this paper, we study the nonexistence of global weak solutions to higher-order time-fractional evolution inequalities with subcritical degeneracy. Using the test function method and some integral estimates, we establish sufficient conditions depending on the parameters of the problems so that global weak solutions cannot exist globally.

Nonexistence of Global Positive Solutions for p-Laplacian Equations with Non-Linear Memory

The Cauchy problem in Rd,d≥1, for a non-local in time p-Laplacian equations is considered. The nonexistence of nontrivial global weak solutions by using the test function method is obtained.

On the global weak solution problem of semilinear generalized Tricomi equations, II

On the global weak solution problem of semilinear generalized Tricomi equations, II

Global existence and eventual smoothness in a 2-D parabolic-elliptic system arising from ion transport networks

The aim of this paper is to investigate a parabolic-elliptic system, which was proposed by Albi et al (2016) to describe the evolution of ion transport networks. Our first result asserts that the corresponding Neumann-Dirichlet initial-boundary value problem possesses a global weak solution for general large data in the two-dimensional setting. Our second result further reveals that such solution at least eventually becomes bounded and smooth, and approaches spatial equilibria in the large time limit.

Remarks on Parabolicity in a One-Dimensional Interdiffusion Model with the Vegard Rule

Until 1948 the interdiffusion theory was based on the Onsager phenomenology, namely thermodynamics of irreversible processes, and a drift was not included. Its main limitation is practical impossibility of the experimental as well as theoretical determination of mobilities (diffusivities) in multicomponent systems (r > 2). After experimental discovery of the drift by Smigelskas and Kirkendall (Trans AIME 171:130–142, 1947), Darken (Trans AIME 175:184–201, 1948) formulated his famous model for the binary system. Consequently, the bi-velocity approach dominates interdiffusion studies (e.g. in more than 500 papers in 2020). In this paper, we consider the diffusional transport in a one-dimensional r-component solid solution. The model is expressed by the nonlinear system of strongly coupled evolution differential equations with initial and nonlinear coupled boundary conditions. We present a non-trivial proof of a theorem called the criterion of parabolicity, which implies the generalized parabolicity condition formulated without a proof in our previous works. This condition is a key in the proofs of our previous theorems on existence, uniqueness and properties of global weak solutions of the differential problem studied. The criterion of parabolicity works if diffusion coefficients are not too dispersed, and it is true in many physical systems. The numerical simulations consistent with real experiments for which our criterion works are given.

Blow-up of a dyadic model with intermittency dependence for the Hall MHD

We derive dyadic models for the magnetohydrodynamics with Hall effect by including the intermittency dimension as a parameter. For such dyadic models, existence of global weak solutions is established. In addition, local strong solution is obtained; while global strong solution is obtained in the case of high intermittency dimension. Moreover, we show that positive solution with large initial data develops blow-up in finite time provided the intermittency dimension is lower than a threshold.

Global conservative weak solutions for the two-component Novikov equation

In this paper, we study the existence of global conservative solutions to a two-component Novikov system. The system is an integrable multi-component extension of the Novikov integrable equation. We develop the method of constructing global conservative solutions of the Camassa–Holm equation due to Bressan and Constantin to a two-component case with genuine nonlinear interactions. Our approach is based on the formulation of Bressan and Constantin and additional estimates relating to the new conservation laws of the system.

Global well-posedness of a Navier–Stokes–Cahn–Hilliard system with chemotaxis and singular potential in 2D

We study a diffuse interface model that describes the dynamics of incompressible two-phase flows with chemotaxis effects. This model also takes into account some significant mechanisms such as active transport and nonlocal interactions of Oono's type. The system under investigation couples the Navier–Stokes equations for the fluid velocity, a convective Cahn–Hilliard equation with physically relevant singular potential for the phase-field variable and an advection-diffusion-reaction equation for the nutrient density. For the initial boundary value problem in a smooth bounded domain Ω ⊂ R 2 , we first prove the existence and uniqueness of global strong solutions that are strictly separated from the pure states ±1 over time. Then we prove the continuous dependence with respect to the initial data and source term for the strong solution in energy norms. Finally, we show the propagation of regularity for global weak solutions.

Blowup criterion of classical solutions for a parabolic-elliptic system in space dimension 3

This paper is concerned with a parabolic-elliptic system, which was originally proposed to model the evolution of biological transport networks. Recent results show that the corresponding initial-boundary value problem possesses a global weak solution, which, in particular, is also classical in the one and two dimensional cases. In this work, we establish a Serrin-type blowup criterion for classical solutions in the three dimensional setting.

Nonexistence of solutions for quasilinear hyperbolic inequalities

In this paper, we study the Cauchy problems for quasilinear hyperbolic inequalities with nonlocal singular source term and prove the nonexistence of global weak solutions in the homogeneous and nonhomogeneous cases by the test function method.

Existence of weak solutions to multiphase Cahn–Hilliard–Darcy and Cahn–Hilliard–Brinkman models for stratified tumor growth with chemotaxis and general source terms

We investigate a multiphase Cahn–Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stratified tumor exhibiting several layers of tissue (e.g., proliferating, quiescent and necrotic tissue) more precisely. Our model consists of a convective Cahn–Hilliard type equation to describe the tumor evolution, a velocity equation for the associated volume-averaged velocity field, and a convective reaction-diffusion type equation to describe the density of the chemical species. The velocity equation is either represented by Darcy’s law or by the Brinkman equation. We first construct a global weak solution of the multiphase Cahn–Hilliard–Brinkman model. After that, we show that such weak solutions of this system converge to a weak solution of the multiphase Cahn–Hilliard–Darcy system as the viscosities tend to zero in some suitable sense. This means that the existence of a global weak solution to the Cahn–Hilliard–Darcy system is also established.

A convective model for poro-elastodynamics with damage and fluid flow towards Earth lithosphere modelling

Devised towards geophysical applications for various processes in the lithosphere or the crust, a model of poro-elastodynamics with inelastic strains and other internal variables like damage (aging) and porosity as well as with diffusion of water is formulated fully in the Eulerian setting. Concepts of gradient of the total strain rate as well as the additive splitting of the total strain rate are used while eliminating the displacement from the formulation. It relies on that the elastic strain is small while only the inelastic and the total strains can be large. The energetics behind this model is derived and used for analysis as far as the existence of global weak energy-conserving solutions concerns. By this way, the model of Lyakhovsky et al. (Appl Geophys 171:3099–3123, 2014; J Mech Phys Solids 59:1752–1776, 2011) is completed to make it mechanically consistent and amenable for analysis.

Small-signal solutions of a two-dimensional doubly degenerate taxis system modeling bacterial motion in nutrient-poor environments

The doubly degenerate nutrient taxis model ut=∇⋅(uv∇u)−∇⋅(u2v∇v)+ℓuv,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0,is considered in smoothly bounded convex subdomains of the plane, with ℓ≥0. It is shown that for any p>2 and each fixed nonnegative u0∈W1,∞(Ω), a smallness condition exclusively involving v0 can be identified as sufficient to ensure that an associated no-flux type initial–boundary value problem with (u,v)|t=0=(u0,v0) admits a global weak solution satisfying esssupt>0‖u(⋅,t)‖Lp(Ω)<∞. The proof relies on the use of an apparently novel class of functional inequalities which provide estimates from below for certain Dirichlet integrals involving possibly degenerate weight functions.

On global solutions to a viscous compressible two-fluid model with unconstrained transition to single-phase flow in three dimensions

We consider the Dirichlet problem for a compressible two-fluid model in multi-dimensions. It consists of the continuity equations for each fluid and the momentum equations for the mixture. This model can be derived from the compressible two-fluid model with equal velocities (Bresch et al., in Arch Rational Mech Anal 196:599–629, 2010) and from a scaling limit of the Vlasov-Fokker-Planck/compressible Navier-Stokes (Mellet and Vasseur, Commun Math Phys 281(3):573–596, 2008) (see also the compressible Oldroyd-B model with stress diffusion (Barrett et al., Commun Math Sci 15:1265–1323, 2017). Another interesting connection is that it is formally the equations of compressible magnetohydrodynamic (MHD) flows without resistivity in two dimensions under the action of vertical magnetic field (Li and Sun, J Differ Equ 267(6): 3827–3851, 2019). Under weak assumptions on the initial data which can be discontinuous, unbounded and large as well as involve transition to pure single-phase points or regions, we show existence of global weak solutions with finite energy. The essential novelty of this work, compared with previous works on the same model, is that transition to each single-phase flow is allowed without any constraints between adiabatic constants or two densities. It means that one of the phases can vanish in a point while the other can persist. The lack of enough regularity for each densities brings up essential difficulties in the two-component pressure compared with the single-phase model, i.e., compressible Navier-Stokes equations. The key points to achieve the main result rely on the variables reduction technique for the pressure function, domain separation, and some new estimates. As a byproduct, we obtain the existence of global weak solutions to the compressible MHD system without resistivity in two dimensions under the action of non-negatively vertical magnetic field, which represents a step forward to the study of the global large solution to the compressible MHD system without resistivity.

Sharp conditions on global existence and blow-up in a degenerate two-species and cross-attraction system

AbstractWe consider a degenerate chemotaxis model with two-species and two-stimuli in dimensiond≥ 3 and find two critical curves intersecting at one point which separate the global existence and blow up of weak solutions to the problem. More precisely, above these curves (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in finite time on and below these curves (i.e. critical and supercritical cases). In addition, the existence or non-existence of minimizers of free energy functional is discussed on the critical curves and the solutions exist globally in time if the size of initial data is small. We also investigate the crossing point between the critical lines in which a refined criteria in terms of the masses is given again to distinguish the dichotomy between global existence and blow up. We also show that the blow ups is simultaneous for both species.

Mathematical analysis of a model describing solute trapping during solidification of binary alloys

In this paper, we present a mathematical analysis to a Cahn-Hilliard/Allen-Cahn system with degenerate mobility that models an isothermal process of solidification of a binary alloy. This model is able to predict an observable phenomenon called solute trapping. The existence of global weak solutions for the system is proved. We approximate the degenerate system and show the convergence of solutions to the approximated non-degenerate problem to a solution of the degenerate one. We also investigated deeply the non-degenerate system by showing the existence of global weak solutions, the existence of global strong solutions in the two-dimensional case, and local strong solutions in the three-dimensional case, as well as, providing conditions to the uniqueness be satisfied.

Quasi-neutral limit of the Isothermal Naiver–Stokes–Poisson with boundary

The quasineutral limit of the isothermal Navier–Stokes–Poisson system is rigorously proved when the combined quasineutral and vanishing viscosity limit is considered in a domain with boundary. The convergence of the global weak solution for Navier–Stokes–Poisson system to the strong solution for incompressible Euler equations is obtained.

Prescribed signal concentration on the boundary: Weak solvability in a chemotaxis-Stokes system with proliferation

We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form where kappa ge 0, mu >0 and, in contrast to the commonly investigated variants of chemotaxis-fluid systems, the signal concentration on the boundary of the domain Omega subset mathbb {R}^N with Nin {2,3} is a prescribed time-independent nonnegative function c_*in C^{2}!left( {{,mathrm{overline{Omega }},}}right) . Making use of the boundedness information entailed by the quadratic decay term of the first equation, we will show that the system above has at least one global weak solution for any suitably regular triplet of initial data.

Existence of solution for a class of heat equation with double criticality

In this paper, we study the following class of quasilinear heat equations{ut−ΔΦu=f(x,u)inΩ,t>0,u=0on∂Ω,t>0,u(x,0)=u0(x)inΩ, where ΔΦu=div(φ(x,|∇φ|)∇φ) and Φ(x,s)=∫0|s|φ(x,σ)σdσ is a generalized N-function. We suppose that Ω⊂RN(N≥2) is a smooth bounded domain that contains two open regions ΩN and Ωp with Ω‾N∩Ω‾p=∅. Under some appropriate conditions, the global existence will be done by combining the Galerkin approximations with the potential well theory. Moreover, the large-time behavior of the global weak solution is analyzed. The main feature of this paper consists that −ΔΦu behaves like −ΔNu on ΩN and −Δpu on Ωp, while the continuous function f:Ω×R→R behaves like eα|s|NN−1 on ΩN and |s|p⁎−2s on Ωp as |s|→∞.