scholarly journals Remark on local solvability of the Cauchy problem for semirelativistic equations

2015 ◽  
Vol 432 (2) ◽  
pp. 744-748 ◽  
Author(s):  
Kazumasa Fujiwara
Keyword(s):  
Cauchy Problem ◽  
Local Solvability ◽  
The Cauchy Problem

On the existence of a continuously differentiable solution to the Cauchy problem for implicit differential equations

2019 ◽  
pp. 376-383
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Initial Point ◽  
Local Solvability ◽  
Time Interval ◽  
Phase Variable ◽  
Existence Of A Solution ◽  
Differentiable Functions ◽  
The Cauchy Problem

We study the question of the existence of a solution to the Cauchy problem for a differential equation unsolved with respect to the derivative of the unknown function. Differential equations generated by twice continuously differentiable mappings are considered. We give an example showing that the assumption of regularity of the mapping at each point of the domain is not enough for the solvability of the Cauchy problem. The concept of uniform regularity for the considered mappings is introduced. It is shown that the assumption of uniform regularity is sufficient for the local solvability of the Cauchy problem for any initial point in the class of continuously differentiable functions. It is shown that if the mapping defining the differential equation is majorized by mappings of a special form, then the solution of the Cauchy problem under consideration can be extended to a given time interval. The case of the Lipschitz dependence of the mapping defining the equation on the phase variable is considered. For this case, estimates of non-extendable solutions of the Cauchy problem are found. The results are compared with known ones. It is shown that under the assumptions of the proved existence theorem, the uniqueness of a solution may fail to hold. We provide examples llustrating the importance of the assumption of uniform regularity.

On the Cauchy problem for implicit differential equations of higher orders

2021 ◽  
pp. 348-362
Author(s):  
Aram V. Arutyunov ◽  
Elena A. Pluzhnikova
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Metric Spaces ◽  
Local Solvability ◽  
Existence Of A Solution ◽  
Nemytskii Operator ◽  
Implicit Differential Equations ◽  
Controlled Systems ◽  
The Cauchy Problem ◽  
The Relationship

The article is devoted to the study of implicit differential equations based on statements about covering mappings of products of metric spaces. First, we consider the system of equations Φ_i (x_i,x_1,x_2,…,x_n )=y_i, i=(1,n,) ̅ where 〖 Φ〗_i: X_i×X_1×… ×X_n→Y_i, y_i∈Y_i, X_i and Y_i are metric spaces, i=(1,n) ̅. It is assumed that the mapping 〖 Φ〗_i is covering in the first argument and Lipschitz in each of the other arguments starting from the second one. Conditions for the solvability of this system and estimates for the distance from an arbitrary given element x_0∈X to the set of solutions are obtained. Next, we obtain an assertion about the action of the Nemytskii operator in spaces of summable functions and establish the relationship between the covering properties of the Nemytskii operator and the covering of the function that generates it. The listed results are applied to the study of a system of implicit differential equations, for which a statement about the local solvability of the Cauchy problem with constraints on the derivative of a solution is proved. Such problems arise, in particular, in models of controlled systems. In the final part of the article, a differential equation of the n-th order not resolved with respect to the highest derivative is studied by similar methods. Conditions for the existence of a solution to the Cauchy problem are obtained.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1287-1293 ◽  
Author(s):  
Zujin Zhang ◽  
Dingxing Zhong ◽  
Shujing Gao ◽  
Shulin Qiu
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
The Cauchy Problem ◽  
Appl Math

In this paper, we consider the Cauchy problem for the 3D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves [S. Rahman, Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure, J. Comput. Appl. Math., 270 (2014), 88-99].

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