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Regular solution

About: Regular solution is a research topic. Over the lifetime, 1065 publications have been published within this topic receiving 23276 citations.


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TL;DR: In this paper, a thermodynamic model based on the regular solution approximation is presented and a formalism, suitable for phases with an arbitrary number of sublattices, is developed, in order to simplify the analytical expressions for the integral Gibbs energy.

843 citations

Journal ArticleDOI
TL;DR: In this paper, a local regular solution for the Navier-Stokes system was constructed for a class of semilinear parabolic equations with dimensionless or scaling invariant norm, where p and q are chosen so that the norm of Lq(0, T; Lp) is dimensionless.

828 citations

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P. S. Negi1
TL;DR: In this paper, it is shown that the exterior Schwarzschild solution itself provides necessary conditions for the types of the density distributions to be considered inside the mass, in order to obtain exact solutions or equations of state compatible with the structure of general relativity.
Abstract: We examine various well known exact solutions available in the literature to investigate the recent criterion obtained in ref. [20] which should be fulfilled by any static and spherically symmetric solution in the state of hydrostatic equilibrium. It is seen that this criterion is fulfilled only by (i) the regular solutions having a vanishing surface density together with the pressure, and (ii) the singular solutions corresponding to a non-vanishing density at the surface of the configuration . On the other hand, the regular solutions corresponding to a non-vanishing surface density do not fulfill this criterion. Based upon this investigation, we point out that the exterior Schwarzschild solution itself provides necessary conditions for the types of the density distributions to be considered inside the mass, in order to obtain exact solutions or equations of state compatible with the structure of general relativity. The regular solutions with finite centre and non-zero surface densities which do not fulfill the criterion [20], in fact, can not meet the requirement of the `actual mass' set up by exterior Schwarzschild solution. The only regular solution which could be possible in this regard is represented by uniform (homogeneous) density distribution. The criterion [20] provides a necessary and sufficient condition for any static and spherical configuration (including core-envelope models) to be compatible with the structure of general relativity. Thus, it may find application to construct the appropriate core-envelope models of stellar objects like neutron stars and may be used to test various equations of state for dense nuclear matter and the models of relativistic stellar structures like star clusters.

791 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that every positive regular solution u(x) is radially symmetric and monotone about some point and therefore assumes the form with constant c = c(n, α) and for some t > 0 and x0 ϵ ℝn.
Abstract: Let n be a positive integer and let 0 < α < n. Consider the integral equation We prove that every positive regular solution u(x) is radially symmetric and monotone about some point and therefore assumes the form with some constant c = c(n, α) and for some t > 0 and x0 ϵ ℝn. This solves an open problem posed by Lieb 12. The technique we use is the method of moving planes in an integral form, which is quite different from those for differential equations. From the point of view of general methodology, this is another interesting part of the paper. Moreover, we show that the family of well-known semilinear partial differential equations is equivalent to our integral equation (0.1), and we thus classify all the solutions of the PDEs. © 2005 Wiley Periodicals, Inc.

781 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20221
202131
202028
201937
201843
201747