Searcharxiv⌕ Search

arXiv subjects

Valentin Lychagin

Publications and source records attributed to Valentin Lychagin.

At least 37 records · Page 2Linked to original sources
arXiv2020

Shock Waves in Euler Flows of Gases

Non-stationary Euler flows of gases are studied. The system of differential equations describing such flows can be represented by means of 2-forms on zero-jet space and we get some exact solutions by means of such a representation. Solutions obtained are multivalued and we provide a method of finding caustics, as well as wave front displacement. The method can be applied to any model of thermodynamic state as well as to any thermodynamic process. We illustrate the method on adiabatic ideal gas flows.

math-ph↗
arXiv2020

Differential invariants for flows of fluids and gases

This paper surveys results found by the authors in the previous papers (see for example, A. Duyunova, V. Lychagin, S. Tychkov, Differential invariants for spherical layer flows of a viscid fluid, Journal of Geometry and Physics, 130, 288-292 (2018)) concerning calculation and classification of symmetry Lie algebra and differential invariants for a variety of fundamental systems in fluid mechanics, including the Euler and Navier-Stokes equations on a plane, three-dimensional space, a sphere, and a spherical layer. In each case, the symmetry Lie algebra is found, followed by a list of generating differential invariants and invariant derivations for the group action. The main extension is a detailed analysis of thermodynamic states, symmetries, and differential invariants. This analysis is based on consideration of Riemannian structure naturally associated with Lagrangian manifolds that represent thermodynamic states. This approach radically changes the description of the thermodynamic part of the symmetry algebra as well as the field of differential invariants.

math-ph↗
arXiv2020

Optimal Thermodynamic Processes For Gases

In this paper, we consider an optimal control problem in equilibrium thermodynamics of gases. Thermodynamic state of the gas is given by a Legendrian submanifold in a contact thermodynamic space. Using Pontryagin's maximum principle we find a thermodynamic process on this submanifold such that the gas maximizes the work functional. For ideal gases, this problem is shown to be integrable in Liouville's sense and its solution is given by means of action-angle variables. For real gases considered as a perturbation of ideal ones, the integrals are given asymptotically.

math-ph↗
arXiv2020

Critical phenomena and singular solutions in non-stationary filtration of real gases

In this paper, we study non-stationary filtration of real gases in porous media. Thermodynamic state of the medium is given by van der Waals state equations. Solutions for non-stationary filtration equation are obtained by means of finite dimensional dynamics. The analysis of phase transitions along the flow in case of isentropic and isenthalpic processes is presented as well as singular properties of solutions obtained are discussed. Domains in the jet space where the dynamics found is an attractor are shown.

math-ph↗
arXiv2019

Non-stationary adiabatic filtration of gases in porous media

A non-stationary isentropic filtration of gases in porous media is considered. Thermodynamics in terms of contact and symplectic geometries is briefly discussed. Algebra of symmetries for the PDE system is found, and the classification of media with respect to admissible symmetries is given. Solution for one class of media is found and the phase transitions for this solution are studied. Keywords: gas filtration, porous media, symmetry algebra, phase

math-ph↗
arXiv2019

Real gas flows issued from a source

Stationary adiabatic flows of real gases issued from a source of given intensity are studied. Thermodynamic states of gases are described by Legendrian or Lagrangian manifolds. Solutions of Euler equations are given implicitly for any equation of state and the behavior of solutions of the Navier-Stokes equations with the viscosity considered as a small parameter is discussed. For different intensities of the source we introduce a small parameter into the Navier-Stokes equation and construct corresponding asymptotic expansions. We consider the most popular model of real gases --- the van der Waals model, and ideal gases as well.

math-ph↗
arXiv2019

Steady filtration of Peng-Robinson gas in a porous medium

Filtration of real gases described by Peng-Robinson equations of state in 3-dimensional space is studied. Thermodynamic states are considered as either Legendrian submanifolds in contact space, or Lagrangian submanifolds in symplectic space. The correspondence between singularities of their projection on the plane of intensives and phase transitions is shown, and coexistence curves in various coordinates are constructed. A method of finding explicit solutions of the Dirichlet boundary problem is provided and the case of a number of sources is discussed in details. The domains corresponding to different phases are shown.

math-ph↗
arXiv2019

Phase transitions in filtration of Redlich-Kwong gases

In this paper we study a 3-dimensional filtration of real gases described by Redlich-Kwong equations of state. Thermodynamical states are considered as Legendrian (Lagrangian) submanifolds in contact (symplectic) space. Connection between singularities of their projection on the space of intensive variables and phase transitions is shown. Explicit formulae for the Dirichlet boundary problem are given and the distribution of phases in space is shown.

math-ph↗
arXiv2019

Phase transitions in filtration of real gases

Steady adiabatic filtration of real gases is studied. Thermodynamical states of real gases are presented by Legendrian surfaces in 5-dimensional thermodynamical contact space. The relation between phase transitions and singularities of projection of the Legendrian surfaces on the plane of intensive variables is shown. The constructive method of finding solutions of the Dirichlet filtration problem together with analysis of critical phenomena is presented. Cases of van der Waals and Peng-Robinson gases are discussed in details.

math-ph↗
arXiv2015

Global Lie-Tresse theorem

We prove a global algebraic version of the Lie-Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.

math.DG↗
arXiv2008

Feedback Differential Invariants

The problem of feedback equivalence for control systems is considered. An algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found.

math.DG↗
arXiv2008

Compatibility, multi-brackets and integrability of systems of PDEs

We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher Jacobi-Mayer brackets, important in the study of evolutionary equations and the integrability problem. We also calculate Spencer delta-cohomology of generalized complete intersections and evaluate the formal functional dimension of the solutions space. The results are applied to establish new integration methods and solve several differential-geometric problems.

math.DG↗
arXiv2007

Differential invariants of the motion group actions

Differential invariants of a (pseudo)group action can vary when restricted to invariant submanifolds (differential equations). The algebra is still governed by the Lie-Tresse theorem, but may change a lot. We describe in details the case of the motion group $O(n)\ltimes\R^n$ acting on the full (unconstraint) jet-space as well as on some invariant equations.

math.DG↗