Global Existence Research Articles

The Cauchy problem for an inviscid and non-diffusive Oldroyd-B model in two dimensions

A two-dimensional inviscid and diffusive Oldroyd-B model was investigated by Elgindi and Rousset (2015) where the global existence and uniqueness of the strong solution were established for arbitrarily large initial data. As pointed out by Bhave et al. (1991), since the diffusion coefficient is significantly smaller than other effects, it is interesting to study the non-diffusive model. In the present work, we obtain the global-in-time existence and uniqueness of the strong solution to the non-diffusive model with small initial data by deriving some uniform regularity estimates and taking vanishing diffusion limits. In addition, the large time behavior of the solution is studied and the optimal time-decay rates for each order of spatial derivatives are obtained. The main challenges focus on the lack of dissipation and regularity effects of the system and on the slower decay in the two-dimensional settings. A combination of the spectral analysis and the Fourier splitting method is adopted.

On the critical exponents for a fractional diffusion-wave equation with a nonlinear memory term in a bounded domain

In this paper, we prove sharp blow-up and global existence results for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain, where the fractional derivative in time is taken in the sense of the Caputo type. Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term in a bounded domain. The proof of blow-up results is based on the eigenfunction method and the asymptotic properties of solutions for an ordinary fractional differential inequality.

Global existence for the stochastic rotation-two-component Camassa–Holm system with nonlinear noise

We study the well-posedness for the stochastic rotation-two-component Camassa–Holm (R2CH) system with the nonlinear noise. We establish the local well-posedness of the stochastic R2CH system by the dispersion–dissipation approximation system and the regularization method. We also prove the global existence of the stochastic R2CH system with a large nonlinear noise.

Global existence of the strong solution to the climate dynamics model with topography effects and phase transformation of water vapor

This study investigates a climate dynamics model that incorporates topographical effects and the phase transformation of water vapor. The system comprises the Navier–Stokes equations, the temperature equation, the specific humidity equation, and the water content equation, all adhering to principles of energy conservation. Applying energy estimation methods, the Helmholtz–Weyl decomposition theorem, and the Brezis–Wainger inequality, we derive high-order a priori estimates for state functions. Subsequently, based on the initial data assumptions V0 ∈ H4(Ω), T0′,q0,mw0∈H2(Ω), we can prove that a strong solution to this system exists globally in time and establish the uniqueness of the global strong solution.

Variational Principles in Quaternionic Analysis with Applications to the Stationary MHD Equations

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder’s fixed point theorem. The advantage of the approach using Schauder’s fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These considerations will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.

Spherically symmetric evolution of self-gravitating massive fields

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including novel positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions.

Global existence, boundedness and asymptotic behavior of classical solutions to a fully parabolic two-species chemotaxis-competition model with singular sensitivity

This paper deals with the following parabolic-parabolic-parabolic chemotaxis system with singular sensitivity and Lotka-Volterra competition kinetics(0.1){ut=Δu−χ1∇⋅(uw∇w)+μ1u(1−u−a1v),t>0,x∈Ω,vt=Δv−χ2∇⋅(vw∇w)+μ2v(1−v−a2u),t>0,x∈Ω,wt=Δw−w+u+v,t>0,x∈Ω,∂u∂ν=∂v∂ν=∂w∂ν=0,t>0,x∈∂Ω,u(0,x)=u0(x),v(0,x)=v0(x),w(0,x)=w0(x),x∈Ω, where Ω⊂RN(N≥2) is a bounded smooth convex domain, and the parameters χ1,χ2,μ1,μ2,a1 and a2 are positive constants. It is shown that the system possesses globally bounded classical solutions under the following conditions χ1,χ2∈(0,12)forN=2,3 or χ1,χ2∈(0,N−2N−1)forN≥4. Moreover, if min⁡{μ1,μ2}>max⁡{χ1,χ2}24, we obtain the uniformly lower bound for w. Finally, when χ1,χ2 are suitably small, it is proved that if 0<a1,a2<1, then the solution (u,v,w) converges to (1−a11−a1a2,1−a21−a1a2,2−a1−a21−a1a2) in L∞ norm as t→∞; if 0<a2<1≤a1, then the solution (u,v,w) converges to (0,1,1) in L∞ norm as t→∞.

Global existence of a pair of coupled Cahn-Hilliard equations with nondegenerate mobility and logarithmic potential

We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qε). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qε). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qε). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces.

Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3

Abstract The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with Chen et al. (Global existence and time–decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307). Under further assumptions that the stationary solution ( ρ s ( x ) , 0 , 0 ) T {\left({\rho }_{s}\left(x),0,0)}^{T} is in a small neighborhood of the constant state ( ρ ¯ , 0 , 0 ) T {\left(\bar{\rho },0,0)}^{T} at infinity, we also obtain the time decay rates of the solution by the combination of the energy method and the linear L p {L}^{p} - L q {L}^{q} decay estimates.

Global well-posedness of Chemotaxis-Navier–Stokes system with refined rough initial data in [formula omitted

We consider the global solvability of the incompressible Chemotaxis-Navier–Stokes system in Rd,d=2,3. We first investigate the global well-posedness of Chemotaxis-Navier–Stokes equations in R2 by involving a double exponential function of the initial velocity field u0 and the initial chemical concentration c0. By only giving the smallness assumption of initial cell density n0 and the horizontal components of velocity field u0h=(u01,u02) in critical Besov spaces, the global existence and uniqueness of the solution for Chemotaxis-Navier–Stokes equations in R3 is also established. The main contribution of this paper is to supply an improved assumption on initial data and remove the smallness condition on initial chemical concentration c0 in both dimensional two and three.

Global well-posedness and decay estimates for the one-dimensional models of blood flow with a general parabolic velocity profile

In this paper, we study the one-dimensional models of blood flow arising from the hemodynamics of aorta, which are derived from the averaging of the Navier–Stokes equations. We establish the global well-posedness and long-time behavior of the viscid 1D models of blood flow in the Sobolev space framework, where a general parabolic velocity profile is considered. Precisely speaking, we prove the global existence of smooth solution when the initial data is sufficiently small. Moreover, by combining the time-weighted energy estimates with the Green function method, we obtain the optimal time decay rate in Lp (2≤p≤∞) norm. In addition, one can see the area-averaged axial velocity decays faster than the cross-sectional area of vessel from the Green function of the linearized system. This observation is essential to study the decay rates of our nonlinear system.

Global energy conservation solution for the [formula omitted] family of Camassa–Holm type equation

In this paper, we investigate the N−abc family of Camassa–Holm type equation with (N+1)-order nonlinearities. This quasi-linear equation is nonlocal with higher order nonlinearities, compared to the Camassa–Holm equation (N=1) and Novikov equation (N=2). Using both the lower order and the higher order energy conservation laws, as well as the characteristic method, we establish the global existence and uniqueness of the Hölder continuous energy weak solution to the N−abc family of Camassa–Holm type equation in the energy space H1(R)×W1,2N(R). Moreover, we show that a very natural and interesting problem is to study how the regularity of solution changes with respect to N. More precisely, we establish Hölder continuous energy weak solutions with the exponent 1−12N. This result precisely shows how the regularity of solution changes with respect to the power of nonlinear wave speed N, and it reveals an intrinsic relation between Camassa–Holm equation (linear wave speed u1), Novikov equation (quadratic wave speed u2) and the N−abc family of Camassa–Holm type equation (wave speed uN).

An integrated multi-polar fuzzy N-soft preference ranking organization method for enrichment of evaluations of the digitization of global economy

Digitization represents the ultimate expression of globalization that has revolutionized every facet of global existence, enhancing connectivity, financial terms, trade opportunities, and public services. To further broaden or fortify this digital realm and advance global progress, various strategies for multi-criteria group decision-making (MCGDM) have been developed. This research article extends the literature study on the sensational MCGDM method known as Preference Ranking Organization Method for Enrichment of Evaluations (PROMETHEE), which shares the burden of selecting the best alternative under many terms and conditions. This article aims to make a positive contribution to the development of the digital world. The main principle of the PROMETHEE technique is to calculate positive and negative flows of the selected options according to the divergence of the alternatives’ scores. It provides us with partial and full rankings of options by determining score degrees, suitable preference functions, and a multi-criteria preference index. To deal with the multi-polarity of the modern world and assign N-ordered levels, this paper proposes the multi-polar fuzzy N-soft (mFNS) PROMETHEE technique. Prior to this, an Analytic Hierarchy Process (AHP) technique is applied to guarantee the credibility of the criterion weights for each criterion. After that, the whole strategy of m-polar fuzzy N-soft PROMETHEE approach to order the selected options is explained, and all the course of action of this remarkable (MCGDM) technique are synchronized in an extensive flowchart, helping us to learn more about the technology keenly. Moreover, the utility of our discussed technology is illustrated by considering two applications of generating electricity through human movement and choosing the best digital currency. The most suitable choice is extracted with the help of the outranked directed graph. The results obtained by mFNS PROMETHEE technique benefits us with the selection of most reliable source to digitized the global economy, in addition, It enables us to rank the alternatives from most preferable to least one, that not only saves time but also produces the better outcomes. Subsequently, the eminence of mFNS PROMETHEE technology is checked by comparability with the prior art. Lastly, the advantages and disadvantages of our proposed technique are supplemented to demonstrate its productivity and shortcomings.

Global existence for parabolic p-Laplace equations with supercritical growth in whole [formula omitted

The focus of this paper is to investigate the global existence and uniqueness of weak solutions for a class of parabolic p-Laplacian equations whose nonlinearity has a supercritical growth.

Global solutions of two‐dimensional inviscid and resistive magnetohydrodynamics system near an equilibrium

We investigate the global existence of classical solutions for two‐dimensional incompressible, inviscid, and resistive magnetohydrodynamics (MHD) system. Assuming that the initial magnetic field is close to an equilibrium state of constant magnetic vorticity, the global existence result could be obtained using a time‐weighted energy estimate.

The global existence and analyticity of a mild solution to the 3D regularized MHD equations

The global existence and analyticity of a mild solution to the 3D regularized MHD equations

The global existence of strong solutions to thermomechanical Cucker-Smale-Stokes equations in the whole domain

The global existence of strong solutions to thermomechanical Cucker-Smale-Stokes equations in the whole domain

The global existence of strong solutions for a non-isothermal ideal gas system

The global existence of strong solutions for a non-isothermal ideal gas system

On a mathematical model for cancer invasion with repellent pH-taxis and nonlocal intraspecific interaction

Starting from a mesoscopic description of cell migration and intraspecific interactions, we obtain by upscaling an effective reaction–diffusion–taxis equation for the cell population density involving spatial nonlocalities in the source term and biasing its motility and growth behavior according to environmental acidity. We prove global existence, uniqueness, and boundedness of a nonnegative solution to a simplified version of the coupled system describing cell and acidity dynamics. A 1D study of pattern formation is performed. Numerical simulations illustrate the qualitative behavior of solutions.

Trend to Equilibrium for Flows With Random Diffusion

Abstract Motivated by the possibility of noise to cure equations of finite-time blowup, the recent work [ 90] by the second and third named authors showed that with quantifiable high probability, random diffusion restores global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as the Patlak–Keller–Segel equation. A question left open is the asymptotic behavior of the solutions, in particular, whether they converge to a steady state. We answer this question by showing that the solutions from [ 90] in the periodic setting converge in Gevrey norm exponentially fast to the uniform distribution as time $t\rightarrow \infty $.