scholarly journals A Well-Posed Boundary Value Problem for Supercritical Flow of Viscoelastic Fluids of Maxwell Type

1990 ◽  
pp. 181-191 ◽  
Author(s):  
Michael Renardy
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Viscoelastic Fluids ◽  
Supercritical Flow ◽  
Well Posed

scholarly journals Wellposedness of Neumann boundary-value problems of space-fractional differential equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Hong Wang ◽  
Danping Yang
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Neumann Boundary Condition ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Fractional Differential ◽  
Neumann Boundary Value Problems ◽  
Well Posed

AbstractFractional differential equation (FDE) provides an accurate description of transport processes that exhibit anomalous diffusion but introduces new mathematical difficulties that have not been encountered in the context of integer-order differential equation. For example, the wellposedness of the Dirichlet boundary-value problem of one-dimensional variable-coefficient FDE is not fully resolved yet. In addition, Neumann boundary-value problem of FDE poses significant challenges, partly due to the fact that different forms of FDE and different types of Neumann boundary condition have been proposed in the literature depending on different applications.We conduct preliminary mathematical analysis of the wellposedness of different Neumann boundary-value problems of the FDEs. We prove that five out of the nine combinations of three different forms of FDEs that are closed by three types of Neumann boundary conditions are well posed and the remaining four do not admit a solution. In particular, for each form of the FDE there is at least one type of Neumann boundary condition such that the corresponding boundary-value problem is well posed, but there is also at least one type of Neumann boundary condition such that the corresponding boundary-value problem is ill posed. This fully demonstrates the subtlety of the study of FDE, and, in particular, the crucial mathematical modeling question: which combination of FDE and fractional Neumann boundary condition, rather than which form of FDE or fractional Neumann boundary condition, should be used and studied in applications.

scholarly journals The Periodic Boundary Value Problem for a Quasilinear Evolution Equation in Besov Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-13
Author(s):  
Yun Wu ◽  
Zhengrong Liu ◽  
Xiang Zhang
Keyword(s):  
Evolution Equation ◽  
Boundary Value Problem ◽  
Besov Space ◽  
Besov Spaces ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Periodic Boundary Value Problem ◽  
Solution Map ◽  
Well Posed

This paper is concerned with the periodic boundary value problem for a quasilinear evolution equation of the following type:∂tu+f(u)∂xu+F(u)=0,x∈T=R/2πZ,t∈R+. Under some conditions, we prove that this equation is locally well-posed in Besov spaceBp,rs(T). Furthermore, we study the continuity of the solution map for this equation inB2,rs(T). Our work improves some earlier results.

A non-homogeneous boundary value problem for the Kuramoto-Sivashinsky equation posed in a finite interval

2020 ◽  
Vol 26 ◽  
pp. 43 ◽  
Author(s):  
Jing Li ◽  
Bing-Yu Zhang ◽  
Zhixiong Zhang
Keyword(s):  
Boundary Value Problem ◽  
Finite Interval ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Solution Map ◽  
Ill Posed ◽  
Initial Boundary Value ◽  
Well Posed ◽  
Sivashinsky Equation

This paper studies the initial boundary value problem (IBVP) for the dispersive Kuramoto-Sivashinsky equation posed in a finite interval (0, L) with non-homogeneous boundary conditions. It is shown that the IBVP is globally well-posed in the space Hs(0, L) for any s > −2 with the initial data in Hs(0, L) and the boundary value data belonging to some appropriate spaces. In addition, the IBVP is demonstrated to be ill-posed in the space Hs(0, L) for any s < −2 in the sense that the corresponding solution map fails to be in C2.

On the analytic solution of the helical equilibrium equation in the MHD approximation

1975 ◽  
Vol 14 (2) ◽  
pp. 305-314 ◽  
Author(s):  
M. L. Woolley
Keyword(s):  
Boundary Value Problem ◽  
Hypergeometric Functions ◽  
Analytic Solution ◽  
Boundary Value ◽  
Elliptic Partial Differential Equation ◽  
Hydrodynamic Pressure ◽  
Additional Information ◽  
Countably Infinite ◽  
Infinite Set ◽  
Well Posed

The second-order elliptic partial differential equation, which describes a class of static ideally conducting magnetohydrodynamic equilibria with helical symmetry, is solved analytically. When the equilibrium is contained within an infinitely long conducting cylinder, the appropriate Dirichiet boundary-value problem may be solved in general in terms of hypergeometric functions. For a countably infinite set of particular cases, these functions are polynomials in the radial co-ordinate; and a solution may be obtained in a closed form. Necessary conditions are given for the existence of the equilibrium, which is described by the simplest of these functions. It is found that the Dirichlet boundary-value problem is not well-posed for these equiilbria; and additional information (equivalent to locating a stationary value of the hydrodynamic pressure) must be provided, in order that the solution be unique.

scholarly journals Lyapunov-type inequality for a Riemann-Liouville type fractional boundary value problem with anti-periodic boundary conditions

2021 ◽  
Vol 40 (4) ◽  
pp. 873-884
Author(s):  
Jagan Mohan Jonnalagadda ◽  
Debananda Basua
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Type Inequality ◽  
Periodic Boundary Conditions ◽  
Fractional Boundary Value Problem ◽  
Liouville Type ◽  
Well Posed ◽  
Lyapunov Type Inequality

In this article, we establish a Lyapunov-type inequality for a two-point Riemann-Liouville type fractional boundary value problem associated with well-posed anti-periodic boundary conditions. As an application, we estimate a lower bound for the eigenvalue of the corresponding fractional eigenvalue problem.

Sign in / Sign up

Export Citation Format

Share Document