AbstractIn this paper, we prove a global well-posedness of the three-dimensional incompressible Navier-Stokes equation under initial data, which belongs to the Lei-Lin-Gevrey space $\begin{array}{}
Z^{-1}_{a,\sigma}
\end{array}$(ℝ3) and if the norm of the initial data in the Lei-Lin space 𝓧−1 is controlled by the viscosity. Moreover, we will show that the norm of this global solution in the Lei-Lin-Gevrey space decays to zero as time approaches to infinity.
Studies of modified Korteweg-de Vries-type equations are of considerable mathematical interest due to the importance of their applications in various branches of mechanics and physics. In this article, using trilinear estimate in Bourgain spaces, we show the local well-posedness of the initial value problem associated with a coupled system consisting of modified Korteweg-de Vries equations for given data. Furthermore, we prove that the unique solution belongs to Gevrey space G σ × G σ in x and G 3 σ × G 3 σ in t. This article is a continuation of recent studies reflected.
AbstractThe Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.
In this paper, the pseudo-differential type operator [Formula: see text] associated with the Bessel type operator [Formula: see text] defined by (2.3) involving the symbol [Formula: see text] whose derivatives satisfy certain growth conditions depending on some increasing sequences, is studied on certain Gevrey spaces. It is shown that the operator [Formula: see text] is a continuous linear map of one Gevrey space into another Gevrey space. A special pseudo-differential type operator called the Gevrey–Hankel type potential is defined and some of its properties are investigated. A variant of [Formula: see text] is also studied.
In this paper, we study spatial analyticity properties of two classes of equations modeling unidirectional waves in nonlinear, dispersive media, namely KdV-type equations and BBM-type equations. The commentary begins with KdV-type equations and the observation that, for a class of such equations, boundedness of a solution suffices to maintain analyticity and so loss of analyticity detects loss of L∞-regularity. For a larger class of KdV-type equations, the same conclusion is valid provided that L∞-boundedness of a solution is replaced by [Formula: see text]-boundedness. It is also shown that these nonlinear dispersive wave equations are amenable to Gevrey-class analysis based on the boundedness of a Sobolev norm. This analysis yields an explicit lower bound on the possible rate of decrease in time of the uniform radius of analyticity of a solution in terms of the assumed Sobolev bound and the Gevrey-norm of the initial data. Attention is then shifted to BBM-type equations. It is shown that, regardless of the strength of the nonlinearity, a solution starting in a Gevrey space remains in this class for all time. Moreover, a lower bound on the possible rate of decrease in time of the uniform analyticity radius has temporal asymptotics that are independent of the degree of the nonlinearity, and so apparently determined in the main by the dispersion.
AbstractFor unbounded operators A1, …, Ad, Gevrey spaces Sλ1, …, λd (A1, …, Ad) of order (λ1, …, λd) are introduced, where the orders λ1, …, λd need not be equal. These extend the notion of Gevrey space defined by Goodman and Wallach where λ1 = … = λd. Several mild conditions on the operators A1, … Ad and the orders λ1, …, λd are presented such that the equality is valid. Examples are included.
Existence and Uniqueness of the Solution to the 3D Navier-Stokes Equations in the Homogeneous Sobolev-Gevrey Space