Global Existence Of Strong Solutions Research Articles

Global existence in critical spaces for non Newtonian compressible viscoelastic flows

We are interested in the multi-dimensional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the “div-curl” structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. In absence of compatible conditions, the new effective flux is introduced, which enables us to capture the dissipation arising from combination of density and deformation tensor. The partial dissipation in non-Newtonian compressible fluids, is weaker than that of classical Navier-Stokes equations.

Global Strong and Weak Solutions to the Initial-Boundary-Value Problem of two-dimensional Compressible MHD System with Large Initial Data and Vacuum

In this paper, we study the barotropic compressible magnetohydrodynamic equations with the shear viscosity being a positive constant and the bulk one being proportional to a power of the density in a general two-dimensional (2D) bounded simply connected domain. For initial density allowed to vanish, we prove that the initial-boundary-value problem of a 2D compressible MHD system admits the global strong and weak solutions without any restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is $\lambda=\rho^\beta$ with $\beta>4/3$. As we known, this is the first result concerning the global existence of strong solutions to the compressible MHD system in general two-dimensional bounded domains with large initial data and vacuum.

Nonlinear Stability of Solutions on the Outer Pressure Problem of Compressible Navier-Stokes System with Temperature-Dependent Heat Conductivity

In this paper, the one-dimensional compressible Navier-Stokes system with outer pressure boundary conditions is investigated. Under some suitable assumptions, we prove that the specific volume and the temperature are bounded from below and above independently of time, and then give the local and global existence of strong solutions. Furthermore, we also obtain the convergence of the global strong solution to a stationary state and the nonlinearly stability of its convergence. It is worth noticing that all the assumptions imposed on the initial data are the same as Takeyuki Nagasawa [Japan.J.Appl.Math.(1988)]. Therefore, our work can be regarded as an improvement of the results of Takeyuki.

Global existence of strong solutions to the multi-dimensional inhomogeneous incompressible MHD equations

This paper is concerned with the Cauchy problem of the multi-dimensional incompressible magnetohydrodynamic equations with inhomogeneous density and fractional dissipation. It is shown that when α+β=1+n2 satisfying 1≤β≤α≤min{3β2,n2,1+n4} and max{n4,n+16}<α for n≥3, the inhomogeneous incompressible MHD equations have a unique global strong solution for the initial data in some Sobolev spaces without requiring small conditions.

Global existence and large time behavior of strong solutions to the Cauchy problem of the 3D heat conducting incompressible MHD flows

In this paper, we consider an initial value problem of inhomogeneous heat conducting magnetohydrodynamic equations in R3. Through some time-weighted a priori estimates, we prove the global existence of strong solution provided that (‖ρ0u0‖L22+‖b0‖L22)(‖∇u0‖L22+‖∇b0‖L22) is suitably small, which extends the corresponding Zhong's results [16,17] to the whole space R3. Furthermore, we also derive the algebraic decay estimates of the solution.

Existence of global strong solutions for a reduced gravity two-and-a-half layer model

In this paper, we consider the existence of global strong solutions to a reduced gravity two-and-a-half layer model in one-dimensional periodic domain. This work is inspired by Constantin-Drivas-Nguyen-Pasqualotto ((2020) [2]). Based on the Bresch-Desjardins entropy and the analysis of the evolution of the active potential, we obtain the continuation criterion and global existence of strong solutions when there is no vacuum. Due to the structure complexity of the reduced gravity two-and-a-half model, we still need to deal with the extra cross terms, which is different from Constantin-Drivas-Nguyen-Pasqualotto ((2020) [2]).

Global Well-Posedness to the 3D Cauchy Problem of Nonhomogeneous Heat Conducting Navier–Stokes Equations with Vacuum and Large Oscillations

We revisit the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic (MHD) equations in $\mathbb{R}^2$. For the initial density allowing vacuum at infinity, we derive the global existence and uniqueness of strong solutions provided that the initial density and the initial magnetic decay not too slowly at infinity. In particular, the initial data can be arbitrarily large. This improves our previous work where the initial density has non-vacuum states at infinity. The result could also be viewed as an extension of the study in L{\"u}-Xu-Zhong for the inhomogeneous case to the full inhomogeneous situation. The method is based on delicate spatial weighted estimates and the structural characteristic of the system under consideration. As a byproduct, we get the global existence of strong solutions to the 2D Cauchy problem for nonhomogeneous heat conducting Navier-Stokes equations with vacuum at infinity.

Global existence of strong solutions for a general incompressible Oldroyd-B system without damping mechanism

In this paper, we consider the three dimensional Cauchy problem for a general incompressible Oldroyd-B system without damping mechanism, and establish the global well-posedness of strong solutions.

The global solvability of the Hall-magnetohydrodynamics system in critical Sobolev spaces

We are concerned with the 3D incompressible Hall-magnetohydrodynamic system (Hall-MHD). Our first aim is to provide the reader with an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita–Kato’s theorem [On the Navier–Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964) 269–315] for the Navier–Stokes equations. Next, we investigate the long-time asymptotics of global solutions of the Hall-MHD system that are in the Fujita–Kato regularity class. A weak-strong uniqueness statement is also proven. Finally, we consider the so-called 2[Formula: see text]D flows for the Hall-MHD system (that is, 3D flows independent of the vertical variable), and establish the global existence of strong solutions, assuming only that the initial magnetic field is small. Our proofs strongly rely on the use of an extended formulation involving the so-called velocity of electron [Formula: see text] and as regards [Formula: see text]D flows, of the auxiliary vector-field [Formula: see text] that comes into play in the total magneto-helicity balance for the Hall-MHD system.

Global solution to the compressible non-isothermal nematic liquid crystal equations with constant heat conductivity and vacuum

This paper considers the initial-boundary value problem of the one-dimensional full compressible nematic liquid crystal flow problem. The initial density is allowed to touch vacuum, and the viscous and heat conductivity coefficients are kept to be positive constants. Global existence of strong solutions is established for any H^{2} initial data in the Lagrangian flow map coordinate, which holds for both vacuum and non-vacuum case. The key difficulty is caused by the lack of the positive lower bound of the density. To overcome such difficulty, it is observed that the ratio of frac{rho _{0(y)}}{rho (t,y)} is proportional to the time integral of the upper bound of temperature and vector director field, along the trajectory. Density weighted Sobolev type inequalities are constructed for both temperature and director field in terms of frac{rho _{0(y)}}{rho (t,y)} and small dependence on their dissipation estimates. Besides this, to deal with cross terms arising due to liquid crystal flow, higher order priori estimates are established by using effective viscous flux.

Global existence of strong solution to the chemotaxis-shallow water system with vacuum in a bounded domain

In this paper, we establish the global existence of strong solution to the chemotaxis-shallow water system with vacuum in a bounded domain Ω⊂R2 provided that the initial data is partially small. We first get a new blow-up criterion and then we use bootstrap argument to obtain the global existence of strong solution.

Rigorous Derivation of Population Cross-Diffusion Systems from Moderately Interacting Particle Systems

Population cross-diffusion systems of Shigesada–Kawasaki–Teramoto type are derived in a mean-field-type limit from stochastic, moderately interacting many-particle systems for multiple population species in the whole space. The diffusion term in the stochastic model depends nonlinearly on the interactions between the individuals, and the drift term is the gradient of the environmental potential. In the first step, the mean-field limit leads to an intermediate nonlocal model. The local cross-diffusion system is derived in the second step in a moderate scaling regime, when the interaction potentials approach the Dirac delta distribution. The global existence of strong solutions to the intermediate and the local diffusion systems is proved for sufficiently small initial data. Furthermore, numerical simulations on the particle level are presented.

A note on global existence of strong solution to the 3D micropolar equations with a damping term

This paper studies the Cauchy problem of the 3D incompressible micropolar equations with a damping term sigma |u|^{beta -1}u (sigma >0, 1le beta <3). It is shown that the strong solutions exist globally for any 1le beta <3.

Global well-posedness and nonlinear stability of a chemotaxis system modelling multiple sclerosis

We consider a system of reaction–diffusion equations including chemotaxis terms and coming out of the modelling of multiple sclerosis. The global existence of strong solutions to this system in any dimension is proved, and it is also shown that the solution is bounded uniformly in time. Finally, a nonlinear stability result is obtained when the chemotaxis term is not too big. We also perform numerical simulations to show the appearance of Turing patterns when the chemotaxis term is large.

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.

Global Well-Posedness for thed-Dimensional Magnetic Bénard Problem without Thermal Diffusion

This paper focuses on the global existence of strong solutions to the magnetic Bénard problem with fractional dissipation and without thermal diffusion inℝdwithd≥3. By using the energy method and the regularization of generalized heat operators, we obtain the global regularity for this model under minimal amount dissipation.

Collective stochastic dynamics of the Cucker-Smale ensemble under uncertain communication

We study the collective dynamics of the Cucker-Smale (C-S) ensemble under random communication. As the effective modeling of the C-S ensemble with infinite size, we introduce a stochastic kinetic C-S equation with a multiplicative white noise. For the proposed stochastic kinetic model with a multiplicative noise, we present a global existence of strong solutions and their asymptotic flocking dynamics, when initial datum is sufficiently regular, and communication weight function has a positive lower bound.

Global strong solution for compressible and radiative MHD flow with density-dependent viscosity and degenerate heat-conductivity in unbounded domains

This paper considers the planar magnetohydrodynamic system with the influence of radiation on the dynamics at high temperature regimes. When the viscosity μ depending on the specific volume of the gas (μ=μ̃1+μ̃2v−α with μ̃1>0,μ̃2≥0) and the heat conductivity κ being a power function of the temperature (κ=κ̃θβ with κ̃>0), the global existence of strong solution with large initial data to the magnetohydrodynamic system is proved in unbounded domains for any α≥0 and β≥0. In particular, the constant coefficients case (μ and κ are positive constants) is also covered in our theorem.

Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three

In this paper, we consider the defocusing Hartree nonlinear Schrödinger equations on T3 with real-valued and even potential V and Fourier multiplier decaying such as |k|−β. By relying on the method of random averaging operators [Deng et al., arXiv:1910.08492 (2019)], we show that there exists β0, which is less than but close to 1, such that for β &amp;gt; β0, we have invariance of the associated Gibbs measure and global existence of strong solutions in its statistical ensemble. In this way, we extend Bourgain’s seminal result [J. Bourgain, J. Math. Pures Appl. 76, 649–702 (1997)], which requires β &amp;gt; 2 in this case.

Global Strong Solutions of the 2D Density-Dependent Incompressible Magnetic Bénard Problem

In this paper, we prove global existence of strong solutions to the 2D density-dependent incompressible magnetic Benard problem in a bounded domain.