scholarly journals Time decay for solutions to the Stokes equations with drift

2018 ◽  
Vol 20 (03) ◽  
pp. 1750046 ◽  
Author(s):  
M. Schonbek ◽  
G. Seregin
Keyword(s):  
Cauchy Problem ◽  
Stokes System ◽  
Stokes Equations ◽  
Time Decay ◽  
Smooth Vector ◽  
Scale Invariant ◽  
The Cauchy Problem ◽  
Divergence Free ◽  
Vector Valued Function ◽  
Vector Valued

In this note, we study the behavior of Lebesgue norms [Formula: see text] of solutions [Formula: see text] to the Cauchy problem for the Stokes system with drift [Formula: see text], which is supposed to be a divergence free smooth vector-valued function satisfying a scale invariant condition.

A note on the fractional calculus in Banach spaces

2005 ◽  
Vol 42 (2) ◽  
pp. 115-130 ◽  
Author(s):  
Hussein A. H. Salem ◽  
A. M. A. El-Sayed ◽  
O. L. Moustafa
Keyword(s):  
Integral Equation ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Existence Result ◽  
Existence Of Weak Solutions ◽  
Weakly Continuous ◽  
The Cauchy Problem ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Fractional Order Integral

O'Regan fixed point theorem is used to establish an existence result for the fractional order integral equation x(t) = g(t)+ ?Ia f(.,x(.))(t), t?[0,1], a ? 0, where the vector-valued function f  is nonlinear weakly-weakly continuous. Moreover, existence of weak solutions to the Cauchy problem  dx/dt = f(t, x (t)), t ? [0,1], x(0) = x0, is obtained as a corollary.

scholarly journals On some perturbations of a stable process and solutions to the Cauchy problem for a class of pseudo-differential equations

2015 ◽  
Vol 7 (1) ◽  
pp. 101-107 ◽  
Author(s):  
M.M. Osypchuk
Keyword(s):  
Cauchy Problem ◽  
Differential Operator ◽  
Euclidean Space ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Stable Process ◽  
Pseudo Differential Operator ◽  
The Cauchy Problem ◽  
Vector Valued Function ◽  
Vector Valued

A fundamental solution for some class of pseudo-differential equations is constructed by the method based on the theory of perturbations. We consider a symmetric $\alpha$-stable process in multidimensional Euclidean space. Its generator $\mathbf{A}$ is a pseudo-differential operator whose symbol is given by $-c|\lambda|^\alpha$, were the constants $\alpha\in(1,2)$ and $c>0$ are fixed. The vector-valued operator $\mathbf{B}$ has the symbol $2ic|\lambda|^{\alpha-2}\lambda$. We construct a fundamental solution of the equation $u_t=(\mathbf{A}+(a(\cdot),\mathbf{B}))u$ with a continuous bounded vector-valued function $a$.

scholarly journals Remarks on the Systems of Semilinear Fractional Rayleigh-Stokes Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Le Dinh Long
Keyword(s):  
Cauchy Problem ◽  
Fourier Analysis ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Newtonian Fluids ◽  
System Of Equations ◽  
Main Technique ◽  
The Cauchy Problem ◽  
The Stability

In this paper, we study the Cauchy problem for a system of Rayleigh-Stokes equations. In this system of equations, we use derivatives in the classical Riemann-Liouville sense. This system has many applications in some non-Newtonian fluids. We obtained results for the existence, uniqueness, and frequency of the solution. We discuss the stability of the solutions and find the solution spaces. Our main technique is to use the Banach mapping theorem combined with some techniques in Fourier analysis.

Regularity properties of Schrödinger equations in vector-valued spaces and applications

2019 ◽  
Vol 31 (1) ◽  
pp. 149-166
Author(s):  
Veli Shakhmurov
Keyword(s):  
Initial Value Problems ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Initial Value ◽  
Physical Systems ◽  
Regularity Properties ◽  
The Cauchy Problem ◽  
Linear And Nonlinear ◽  
Vector Valued Function ◽  
Vector Valued

Abstract In this paper, regularity properties and Strichartz type estimates for solutions of the Cauchy problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Banach space E, in which by choosing E and A, we can obtain numerous classes of initial value problems for Schrödinger equations, which occur in a wide variety of physical systems.

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