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Partial Differential Equations and Mathematical Physics

Mathematical physics is a domain of mathematics analyzing well-posedness of models of physical phenomena. Mostly, laws of nature are formulated in terms of partial differential equations. The study of well-posedness of various problems for partial differential equations includes the question of existence and uniqueness of solutions and also obtaining of maximally complete information concerning qualitative properties of solutions. The modern theory of partial differential equations uses a wide class of methods and interacts with various domains of mathematics including the classical analysis, functional analysis, differential geometry, theory of dynamical systems, and others.

Regularity theory

Regularity theory studies the smoothness (i.e., the presence or absence of singularities) of solutions of nonlinear equations and systems of partial differential equations. A typical situation for scalar elliptic and parabolic equations with smooth data is the absence of singularities (solutions of such equations are smooth functions — this problem was formulated by Hilbert as one of his famous 23 problems). For systems, the situation changes, and solutions of nonlinear elliptic equations may lose smoothness even for smooth data of the problem. Another examples of appearance of singularities of solutions are the formation of shock waves in conservation laws and propagation of singularities of initial values of hyperbolic equations along characteristics. The presence or absence of singular points of a solution is an important characteristic which, as a rule, is related to interesting physical phenomena.

Among the people working in this field are N.D. Filonov and T.N. Shilkin.

Evolution equations

One can treat partial differential equations of evolutionary type as dynamical systems for functions with values in an infinite-dimensional (Banach) space. For that reason, in the study of such equations, the difficulties of the classical theory of dynamical systems intertwine with meaningful problems of topology and functional analysis. The main problems for study are the existence and uniqueness of solutions under various assumptions on the initial values, stability of solutions, and their behavior at long times. In its abstract form, the theory of evolutionary equations can be treated as a part of functional analysis (the theory of semigroups), but, taking into account the particular specifics of some evolutionary problems appearing in mathematical physics, it is possible to essentially `move forward’ the theory (in particular, because particular models of mathematical physics, as a rule, have additional properties, such as conservation laws, symmetries, etc). Problems of evolutionary type include a wide range of models, from the classical wave and heat equations and to one of the most famous models of the modern theory of partial differential equations — the Navier-Stokes system.

Among the people working in this field are N.D. Filonov and T.N. Shilkin.

Spectral theory

Spectral theory of differential operators considers parametric families of differential operators and selects values of parameter for which the corresponding equation has a `particular’ solution, i.e., a solution with some “nonstandard” properties (for example, fast decay at infinity, boundedness, etc). In the finite-dimensional case, the spectrum of a linear operator is the set of its eigenvalues. In the infinite-dimensional case (i.e., for differential operators), the situation becomes really more complicated and interesting; in addition to the point spectrum, an operator may have continuous and residual spectra. The spectrum of a linear operator is its very important characteristic which allows one to describe properties of the operator. This is extremely important for applications since in many applied problems the observer is informed just about the spectrum and not about the operator itself. The spectral theory of differential operators is closely related to such domains of mathematical physics as quantum mechanics and quantum field theory.

Among the people working in this field is N.D. Filonov.