Global Regularity Of Solutions Research Articles

GLOBAL REGULARITY TO THE NAVIER-STOKES EQUATIONS FOR A CLASS OF LARGE INITIAL DATA

In [5], Chemin, Gallagher and Paicu proved the global regularity of solutions to the classical Navier-Stokes equations with a class of large initial data on T2 × R. This data varies slowly in vertical variable and has a norm which blows up as the small parameter ( represented by ǫ in the paper) tends to zero. However, to the best of our knowledge, the result is still unclear for the whole spaces R3. In this paper, we consider the generalized Navier-Stokes equations on Rn(n ≥ 3): ∂tu + u · ∇u + Dsu + ∇P = 0, div u = 0. For some suitable number s, we prove that the Cauchy problem with initial data of the form u0ǫ(x) = (v0h(xǫ), ǫ−1v0n(xǫ))T , xǫ = (xh, ǫxn)T , is globally well-posed for all small ǫ > 0, provided that the initial velocity profile v0 is analytic in xn and certain norm of v0 is sufficiently small but independent of ǫ. In particular, our result is true for the n-dimensional classical Navier-Stokes equations with n ≥ 4 and the fractional Navier-Stokes equations with 1 ≤ s < 2 in 3D.

Global regularity for d-dimensional micropolar equations with fractional dissipation

ABSTRACTThis paper is devoted to the global in time existence of classical solutions to the d-Dimensional (dD) micropolar equations with fractional dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. It remains unknown whether or not smooth solutions of the classical 3D micropolar equations can develop finite-time singularities. The purpose here is to explore the global regularity of solutions for dD micropolar equations under the smallest amount of dissipation. We establish the global regularity for two important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ the Besov space techniques.

Dynamics for the diffusive Leslie-Gower model with double free boundaries

In this paper we investigate a free boundary problem for the diffusive Leslie-Gower prey-predator model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species. We first prove the existence, uniqueness and regularity of global solution. Then provide a spreading-vanishing dichotomy, namely the predator species either successfully spreads to infinity as $t\to\infty$ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run. The long time behavior of $(u,v)$ and criteria for spreading and vanishing are also obtained. Because the term $v/u$ (which appears in the second equation) may be unbounded when $u$ nears zero, it will bring some difficulties for our study.

Higher regularity of solutions of singular parabolic equations with variable nonlinearity

ABSTRACTWe study the global regularity of solutions of the homogeneous Dirichlet problem for the parabolic equation with variable nonlinearity where p(x, t), are given functions of their arguments, and . Conditions on the data are found that guarantee the existence of a unique strong solution such that and . It is shown that if with , p and are Hölder-continuous in , and , then for every strong solution with any .

Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat

A three-dimensional thermo-visco-elastic system for Kelvin-Voigt type material at small strains is considered. The system involves nonlinear temperature-dependent specific heat relevant in the limit of low temperature range. The existence of a unique global regular solution is proved without small data assumptions. The proof consists of two parts. First the existence of a local in time solution is proved by the Banach successive approximations method. Then a lower bound on temperature and global a priori estimates are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm. Such estimates allow to extend the local solution step by step in time. The paper generalizes the results of the previous author's publication in SIAM J. Math. Anal. 45, No. 4 (2013), pp. 1997–2045.

On global regular solutions to the mhd equations in a smooth toroidal domain

On global regular solutions to the mhd equations in a smooth toroidal domain

The regularization of solution for the coupled Navier-Stokes and Maxwell equations

The purpose of this paper is to build the existence of time-spatial global regular solution to the coupled Navier-Stokes and Maxwell equations.

On global regular solutions to magnetohydrodynamics in axi-symmetric domains

We consider mhd equations in three-dimensional axially symmetric domains under the Navier boundary conditions for both velocity and magnetic fields. We prove the existence of global, regular axi-symmetric solutions and examine their stability in the class of general solutions to the mhd system. As a consequence, we show the existence of global, regular solutions to the mhd system which are close in suitable norms to axi-symmetric solutions.

Optimal Credit Investment with Borrowing Costs

We consider the portfolio decision problem of a risky investor. The investor borrows at a rate higher than his lending rate and invests in a risky bond whose market price is correlated with the credit quality of the investor. By viewing the concave drift of the wealth process as a continuous function of the admissible control, we characterize the optimal strategy in terms of a relation between a critical borrowing threshold and solutions of a system of first-order conditions. We analyze the nonlinear dynamic programming equation and prove the singular growth of its coefficients. Using a truncation technique relying on the locally Lipschitz continuity of the optimal strategy, we remove the singularity and show the existence and uniqueness of a global regular solution. Our explicit characterization of the strategy has direct financial implications: it indicates that the investor purchases a high number of bond shares when his borrowing costs are low and the bond sufficiently safe, and reduces the size of his long position or even sells short when his financing costs are high or the bond very risky.

Global Regular Solutions to One Problem of Saut-Temam for the 3D Zakharov–Kuznetsov Equation

An initial-boundary value problem for the 3D Zakharov-Kuznetsov equation posed on an unbounded domain is considered. Existence and uniqueness of a global regular solution as well as exponential decay of the $H^2$-norm for small initial data are proven.

Global regular solutions for the 3D Kawahara equation posed on unbounded domains

An initial boundary value problem for the 3D Kawahara equation posed on a channel-type domain was considered. The existence and uniqueness results for global regular solutions as well as exponential decay of small solutions in the H2-norm were established.

Stability of two-dimensional solutions to the Navier–Stokes equations in cylindrical domains under Navier boundary conditions

The Navier–Stokes motions in a cylindrical domain with Navier boundary conditions are considered. First the existence of global regular two-dimensional solutions is proved. The solutions are such that norms bounded with respect to time are controlled by the same constant for all t∈R+. Assuming that the initial velocity and the external force are sufficiently close to the initial velocity and the external force of a two-dimensional solution, we prove existence of global three-dimensional solutions which remain close to the two-dimensional solution for all time. In this sense we have stability of two-dimensional solutions. Thanks to the Navier boundary conditions the nonlinear term in the two-dimensional Navier–Stokes equations does not influence the energy estimate. This implies that the global two-dimensional solution is proved without any structural restrictions on the external force, initial data or viscosity.

Spherically Symmetric Barotropic Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous Initial Data

This paper is concerned with the exterior problem and the initial boundary value problem for the spherically symmetric barotropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and discontinuous initial data. For the exterior problem and the initial boundary value problem, we prove that there exists a unique global piecewise regular solution for piecewise regular initial density with arbitrarily large jump discontinuity. Moreover, we show that the jump of density decays exponentially in time and the piecewise regular solution tends to the equilibrium state exponentially as time tends to infinity.

Global regular solutions to two-dimensional thermoviscoelasticity

A two-dimensional thermoviscoelastic system of Kelvin-Voigt type with strong dependence on temperature is considered. The existence and uniqueness of a global regular solution is proved without small data assumptions. The global existence is proved in two steps. First global a priori estimate is derived applying the theory of anisotropic Sobolev spaces with a mixed norm. Then local existence, proved by the method of successive approximations for a sufficiently small time interval, is extended step by step in time. By two-dimensional solution we mean that all its quantities depend on two space variables only.

Global regular solutions for the 3D Zakharov-Kuznetsov equation posed on unbounded domains

An initial-boundary value problem for the 3D Zakharov-Kuznetsov equation posed on unbounded domains is considered. Existence and uniqueness of a global regular solution as well as exponential decay of the H2-norm for small initial data are proven.

The Kawahara equation on bounded intervals and on a half-line

We consider an initial–boundary value problem for the Kawahara equation posed on bounded intervals and on a half-line. We prove the existence and uniqueness of global regular and weak solutions on bounded intervals (0,L) as well as a priori estimates which do not depend on L and prove that regular and weak solutions on bounded intervals converge to regular and weak solutions on the positive half-line as L→∞.

Regularity of stagnation-point form solutions of the two-dimensional Euler equations

A class of semi-bounded solutions of the two-dimensional incompressible Euler equations, satisfying either periodic or Dirichlet boundary conditions, is examined. For smooth initial data, new blowup criteria in terms of the initial concavity profile is presented and the effects that the boundary conditions have on the global regularity of solutions is discussed. In particular, by deriving a formula for a general solution along Lagrangian trajectories, we describe how periodicity can prevent blow-up. This is in opposition to Dirichlet boundary conditions which, as we will show, allow for the formation of singularities in finite time. Lastly, regularity of solutions arising from non-smooth initial data is briefly discussed.

Existence results for viscous polytropic fluids with degenerate viscosity coefficients and vacuum

In this paper, we considered the isentropic Navier–Stokes equations for compressible fluids with density-dependent viscosities in R3. These systems come from the Boltzmann equations through the Chapman–Enskog expansion to the second order, cf. [17], and are degenerate when vacuum appears. We firstly establish the existence of the unique local regular solution (see Definition 1.1 or [11]) when the initial data are arbitrarily large with vacuum at least appearing in the far field. Moreover it is interesting to show that we couldn't obtain any global regular solution satisfying that the L∞ norm of u decays to zero as time t goes to infinity.

A note on global well-posedness of solutions to boussinesq equations with fractional dissipation

The goal of this paper is to consider the global well-posedness to n-dimensional (n ≥ 3) Boussinesq equations with fractional dissipation. More precisely, it is proved that there exists a unique global regular solution to the Boussinesq equations provided the real parameter α satisfies α ≥ 12+n4

GLOBAL SOLUTIONS TO THE NONLINEAR HYPERBOLIC PROBLEM

We prove the existence and uniqueness of global regular solutions to the mixed problem for the nonlinear hyperbolic equation with nonlinear damping. Utt a (u) u + I u t IP Ut = f (x .t ) in (O, 1) x (O, T ) = Q, u (o.. ) = u (l,t ) = 0, u(x,O) = Uo(x), ut (x,O) = ui (x), Where a (u ) ~ ao > 0, p > 1.No restrictions on a size of uo, ui, f are imposed. It is well-known that quasilinear hyperbolic equations, generally speaking, do not have regular solutions for all t >0_Their solutions can blow up at a finite period of time. See examples of such singularities in [1, 3]. On the other hand,it was observed that adding a linear damping to the nonlinear hyperbolic equations one can expect the existence of global regular solutions provided initial conditions and right-hand side have sufficiently small appropriate norms, [2]. Moreover, in [3] was shown that the presence of the nonlinear damping allows to prove the existence of regular solutions for the equation without restrictions on a size ofthe initial data and f. Later, using the idea of [3], we proved in [4] the existence of regular solutions for the damped Carrier equation. Utt -M ~u(t)12)L1U + alu, IP u, = f without smallness conditions for the initial data and f . • Supported by CNP-Brazil as Visiting Profesor as the State University ofMaringa. Here, we continue to exploit this idea and consider the following nonlinear mixed problem.