Cauchy problem in a scale of banach spaces and its application to the shallow water theory justification

1976 ◽  
pp. 426-437 ◽  
Author(s):  
L. V. Ovsjannikov
Keyword(s):  
Cauchy Problem ◽  
Shallow Water ◽  
Banach Spaces ◽  
Shallow Water Theory ◽  
Scale Of Banach Spaces

scholarly journals An Application Of The Theory Of Scale Of Banach Spaces

2015 ◽  
Vol 29 (1) ◽  
pp. 51-59
Author(s):  
Łukasz Dawidowski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Banach Spaces ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem ◽  
Selection Of

AbstractThe abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

scholarly journals The Cauchy problem in scale of Banach spaces with deviating variables

2021 ◽  
Vol 22 (1) ◽  
pp. 219-230
Author(s):  
Nguyen Bich Huy ◽  
Pham Van Hien
Keyword(s):  
Cauchy Problem ◽  
Banach Spaces ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem

scholarly journals A fixed point theorem and an application for the Cauchy problem in the scale of Banach spaces

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4387-4398 ◽  
Author(s):  
Vo Tri ◽  
Erdal Karapinar
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Convex Space ◽  
Locally Convex Space ◽  
Normed Spaces ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem ◽  
Locally Convex

The main aim of this paper is to prove the existence of the fixed point of the sum of two operators in setting of the cone-normed spaces with the values of cone-norm belonging to an ordered locally convex space. We apply this result to prove the existence of global solution of the Cauchy problem with perturbation of the form (x?(t) = f[t,x(t)] + g[t,x(t)], t ? [0,?), x(0) = x0? F1, in a scale of Banach spaces {(Fs,||.||) : s ? (0, 1]}.

scholarly journals A second-order Cauchy problem in a scale of Banach spaces and application to Kirchhoff equations

2004 ◽  
Vol 206 (1) ◽  
pp. 253-264 ◽  
Author(s):  
Nguyen Bich Huy ◽  
Nguyen An Sum ◽  
Nguyen Anh Tuan
Keyword(s):  
Cauchy Problem ◽  
Banach Spaces ◽  
Second Order ◽  
Kirchhoff Equations ◽  
Scale Of Banach Spaces

Cauchy problem in a scale of Banach spaces

2013 ◽  
Vol 281 (1) ◽  
pp. 3-11 ◽  
Author(s):  
L. V. Ovsyannikov
Keyword(s):  
Cauchy Problem ◽  
Banach Spaces ◽  
Scale Of Banach Spaces

scholarly journals A Shallow-Water Theory for the Sun's Active Longitudes

2005 ◽  
Vol 635 (2) ◽  
pp. L193-L196 ◽  
Author(s):  
Mausumi Dikpati ◽  
Peter A. Gilman
Keyword(s):  
Shallow Water ◽  
Shallow Water Theory

Hodograph Method for Solving the Problem of Shallow Water Under a Solid Lid

2021 ◽  
pp. 15-24
Author(s):  
Tatiana F. Dolgikh
Keyword(s):  
Cauchy Problem ◽  
Green Function ◽  
Differential Equations ◽  
Shallow Water ◽  
Ordinary Differential Equations ◽  
Riemann Invariants ◽  
Hodograph Method ◽  
First Order ◽  
The Cauchy Problem ◽  
Derivatives Of

One of the mathematical models describing the behavior of two horizontally infinite adjoining layers of an ideal incompressible liquid under a solid cover moving at different speeds is investigated. At a large difference in the layer velocities, the Kelvin-Helmholtz instability occurs, which leads to a distortion of the interface. At the initial point in time, the interface is not necessarily flat. From a mathematical point of view, the behavior of the liquid layers is described by a system of four quasilinear equations, either hyperbolic or elliptic, in partial derivatives of the first order. Some type shallow water equations are used to construct the model. In the simple version of the model considered in this paper, in the spatially one-dimensional case, the unknowns are the boundary between the liquid layers h(x,t) and the difference in their velocities γ(x,t). The main attention is paid to the case of elliptic equations when |h|<1 and γ>1. An evolutionary Cauchy problem with arbitrary sufficiently smooth initial data is set for the system of equations. The explicit dependence of the Riemann invariants on the initial variables of the problem is indicated. To solve the Cauchy problem formulated in terms of Riemann invariants, a variant of the hodograph method based on a certain conservation law is used. This method allows us to convert a system of two quasilinear partial differential equations of the first order to a single linear partial differential equation of the second order with variable coefficients. For a linear equation, the Riemann-Green function is specified, which is used to construct a two-parameter implicit solution to the original problem. The explicit solution of the problem is constructed on the level lines (isochrons) of the implicit solution by solving a certain Cauchy problem for a system of ordinary differential equations. As a result, the original Cauchy problem in partial derivatives of the first order is transformed to the Cauchy problem for a system of ordinary differential equations, which is solved by numerical methods. Due to the bulkiness of the expression for the Riemann-Green function, some asymptotic approximation of the problem is considered, and the results of calculations, and their analysis are presented.

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