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The main difficulty with nonlinear partial differential equations is that many of them are not solved exactly. For practical purposes, such equations are solved numerically, and the questions of the existence and uniqueness of their solutions become problems over which scientists have been struggling for decades, and sometimes centuries. One of these problems—Navier-Stokes existence and smoothness—was included in the famous list of Millennium Prize problems: The Clay Mathematical Institute in the U.S. offers a prize of $1 million for solving any of these problems.
Any partial differential equation is defined in a certain area, e.g., on a plane or in a sphere, or in space. Usually, it is possible to find a solution to such equations in a small neighborhood of a point, i.e., a local solution. But it may remain unclear.
RUDN University Mathematical Institute mathematician Andrei Muravnik used the method of inequalities. He generalized the existing theorems to the quasilinear case that arises in the study of the KPZ-type equations. The conditions obtained not only limit the set of possible solutions to the KPZ-type equations, but are also are necessary for the solvability of problems that arise in practice. In particular, these results help in solving the problems of surface growth when modeling the behavior of polymers, and can also be used in the theory of neural networks.
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Additional resources
A. B. Muravnik. On absence of global solutions of quasilinear differential-convolutional inequalities, Complex Variables and Elliptic Equations (2019).
DOI: 10.1080/17476933.2019.1639049
Source: Phys.Org
