The problem discussed is the Navier-Stokes problem (NSP) in \(\mathbb{R}^3\). Uniqueness of its solution is proved in a suitable space \(X\). No smallness assumptions are used in the proof. Existence of the solution in \(X\) is proved for \(t\in [0,T]\), where \(T>0\) is sufficiently small. Existence of the solution in \(X\) is proved for \(t\in [0,\infty)\) if some a priori estimate of the solution holds.
In this paper, we characterize the centre of dense irreducible subalgebras of compact elementary operators that are spectrally bounded. We show that the centre is a unital, irreducible and commutative \(C^{*}\)-subalgebra. Furthermore, the supports from the centre are orthogonal and the intersection of a nonzero ideal with the centre is non-zero.
By applying Opoola differential operator, in this article, two new subclasses \(\mathcal{M}_{\mathcal{H},\sigma}^{\mu,\beta}(m,\psi,k,\tau)\) and \(\mathcal{M}_{\mathcal{H},\sigma}^{\mu,\beta}(m,\xi,k,\tau)\) of bi-univalent functions class \(\mathcal{H}\) defined in \(\bigtriangledown\) are introduced and investigated. The estimates on the coefficients \(|l_2|\) and \(|l_3|\) for functions of the classes are also obtained.
Here we study the existence of solutions of a nonlocal two-point, with parameters, boundary value problem of a first order nonlinear differential equation. The maximal and minimal solutions will be proved. The continuous dependence of the unique solution on the parameters of the nonlocal condition will be proved. The anti-periodic boundary value problem will be considered as an application.
This paper presents an existence theorem of the solutions for a boundary value problem of fractional order differential equations with integral boundary conditions, by using measure of noncompactness combined with Mönch fixed point theorem. An example is furnished to illustrate the validity of our outcomes.
Recently, asymptotic estimates for the unimodular Fourier multipliers \(e^{i\mu(D)}\) have been studied for the function \(\alpha\)-modulation space. In this paper, using the almost orthogonality of projections and some techniques on oscillating integrals, we obtain asymptotic estimates for the unimodular Fourier multiplier \(e^{it(I-\Delta)^{\frac{\beta}{2}}}\) on the \(\alpha\)-modulation space. For an application, we give the asymptotic estimate of the solution for the Klein-Gordon equation with initial data in a \(\alpha\)-modulation space. We also obtain a quantitative form about the solution to the nonlinear Klein-Gordon equation.
In this work, we consider a plate equation with nonlinear source and partially hinged boundary conditions. Our goal is to show analytically that the solution blows up in finite time. The background of the problem comes from the modeling of the downward displacement of suspension bridge using a thin rectangular plate. The result in the article shows that in the present of fractional damping and a nonlinear source such as the earthquake shocks, the suspension bridge is bound to collapse in finite time.
Using the Krasnoselskii’s fixed point theorem and the contraction mapping principle we give sufficient conditions for the existence and uniqueness of solutions for initial value problems for delay fractional differential equations with the mixed Riemann-Liouville and Caputo fractional derivatives. At the end, an example is given to illustrate our main results.
Ation frical contact problem between a piezoelectric body and a deformable conductive foundation is numerically studied in this paper. The process is quasistatic and the material’s behavior is modelled with an electro-viscoelastic constitutive law. Contact is described with the normal compliance condition, a version of Coulomb’s law of dry friction, and a regularized electrical conductivity condition. A fully discrete scheme is introduced to solve the problem. Under certain solution regularity assumptions, we derive an optimal order error estimate. Some numerical simulations are included to show the performance of the method.
Using the technique of differential subordination, we here, obtain certain sufficient conditions for starlike and convex functions. In most of the results obtained here, the region of variability of the differential operators implying starlikeness and convexity of analytic functions has been extended. The extended regions of the operators have been shown pictorially.
