scholarly journals AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION

2013 ◽  
Vol 28 (22n23) ◽  
pp. 1340015 ◽  
Author(s):  
DAVID HILDITCH

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace–Fourier method for analyzing the initial boundary value problem. Finally, I state how these notions extend to systems that are first-order in time and second-order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

Author(s):  
R. W. Dickey

SynopsisThe existence of a classical solution to the initial boundary value problem for a semi-infinite extensible string is proved. The result is obtained by using a Galerkin procedure on a semi-infinite interval.


Author(s):  
Alexander N. Polkovnikov

We consider initial boundary value problem for uniformly 2-parabolic differential operator of second order in cylinder domain in Rn with non-coercive boundary conditions. In this case there is a loss of smoothness of the solution in Sobolev type spaces compared with the coercive situation. Using by Faedo-Galerkin method we prove that problem has unique solution in special Bochner space


Author(s):  
Sharif E. Guseynov ◽  
Ruslans Aleksejevs ◽  
Jekaterina V. Aleksejeva

In the present paper, we propose an analytical approach for solving the 3D unsteady-state boundary-value problem for the second-order parabolic equation with the second and third types boundary conditions in two-layer rectangular parallelepipedic domain.


2018 ◽  
Vol 15 (02) ◽  
pp. 349-374 ◽  
Author(s):  
Elena Rossi

We consider four definitions of solution to the initial-boundary value problem (IBVP) for a scalar balance laws in several space dimensions. These definitions are extended to the same most general framework and then compared. The first aim of this paper is to detail differences and analogies among them. We focus then on the ways the boundary conditions are fulfilled according to each definition, providing also connections among these various modes. The main result is the proof of the equivalence among the presented definitions of solution.


2015 ◽  
Vol 12 (02) ◽  
pp. 221-248 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo

The Ostrovsky–Hunter equation provides a model for small-amplitude long waves in a rotating fluid of finite depth. It is a nonlinear evolution equation. Here the well-posedness of bounded solutions for a non-homogeneous initial-boundary value problem associated with this equation is studied.


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