Equations Of Hydrodynamic Type Research Articles

A priori bounds for Gevrey--Sobolev norms of space-periodic three-dimensional solutions to equations of hydrodynamic type

We present a technique for derivation of a priori bounds for Gevrey--Sobolev norms of space-periodic three-dimensional solutions to evolutionary partial differential equations of hydrodynamic type. It involves a transformation of the flow velocity in the Fourier space, which introduces a feedback between the index of the norm and the norm of the transformed solution, and results in emergence of a mildly dissipative term. We illustrate the technique, using it to derive finite-time bounds for Gevrey--Sobolev norms of solutions to the Euler and inviscid Burgers equations, and global-in-time bounds for the Voigt-type regularizations of the Euler and Navier--Stokes equation (assuming that the respective norm of the initial condition is bounded). The boundedness of the norms implies analyticity of the solutions in space.

Study of the blow-up of solutions of systems of equations of hydrodynamic type

We study the initial boundary-value problem for three-dimensional systems of equations of pseudoparabolic type. The system is similar to the Oskolkov system, but differs from it by the presence of a source of arbitrary order and of a nonlinearity multiplying the highest derivative with respect to time. The local (in time) solvability of the problem is proved in the weak generalized sense. Sufficient conditions for the global (in time) solvability are obtained. We find estimates for the existence time of the solution and for the initial function associated with the blow-up of the solution in finite time.

Von mises- and crocco-type hydrodynamical transformations: Order reduction of nonlinear equations, construction of Bäcklund transformations and of new integrable equations

Broad classes of nonlinear equations of mathematical physics are described that admit order reduction by applying the von Mises transformation (with the unknown function used as a new independent variable and with a suitable partial derivative used as a new dependent variable) and by applying the Crocco transformation (with the first and second partial derivatives used as new independent and dependent variables, respectively). Associated Backlund transformations are constructed that connect evolution equations of general form (their special cases include Burgers, Korteweg-de Vries, and Harry Dym type equations and many other nonlinear equations of mathematical physics). Transformations are indicated that reduce the order of hydrodynamic-type equations of higher orders. The generalized Calogero equation and a number of other new integrable nonlinear equations, reducible to linear equations, are considered.

Integrable string models with constant torsion in terms of chiral invariants of SU(n), SO(n), SP(n groups

We used the invariant local chiral currents of principal chiral models for SU(n), SO(n), SP(n) groups to construct new integrable string equations of hydrodynamic type on the Riemann space of the chiral primitive invariant currents and on the chiral nonprimitive Casimir operators as Hamiltonians.

Systems of hydrodynamic type equations: Exact solutions, transformations, and nonlinear stability

Systems of hydrodynamic type equations: Exact solutions, transformations, and nonlinear stability

Abstract Kelvin–Noether Theorems for Lie Group Extensions

We present an abstract Kelvin–Noether theorem for geodesic equations on abelian Lie group extensions with right invariant metrics and we apply it to equations of hydrodynamical type. Another Kelvin–Noether theorem for a class of central extensions of semidirect products is shown.

Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups

We considered two types of string models: on the Riemmann space of string coordinates with null torsion and on the Riemman-Cartan space of string coordinates with constant torsion. We used the hydrodynamic approach of Dubrovin, Novikov to integrable systems and Dubrovin solutions of WDVV associativity equation to construct new integrable string equations of hydrodynamic type on the torsionless Riemmann space of chiral currents in first case. We used the invariant local chiral currents of principal chiral models for SU(n), SO(n), SP(n) groups to construct new integrable string equations of hydrodynamic type on the Riemmann space of the chiral primitive invariant currents and on the chiral non- primitive Casimir operators as Hamiltonians in second case. We also used Pohlmeyer tensor nonlocal currents to construct new nonlocal string equation.

Supersymmetric version of a hydrodynamic system in Riemann invariants and its solutions

In this paper, a supersymmetric extension of a system of hydrodynamic-type equations involving Riemann invariants is formulated in terms of a superspace and superfield formalism. The symmetry properties of both the classical and supersymmetric versions of this hydrodynamical model are analyzed through the use of group-theoretical methods applied to partial differential equations involving both bosonic and fermionic variables. More specifically, we compute the Lie superalgebras of both models and perform classifications of their respective subalgebras. A systematic use of the subalgebra structures allows us to construct several classes of invariant solutions, including traveling waves, centered waves, and solutions involving monomials, exponentials, and radicals.

A Geometric Study of the Dispersionless Boussinesq Type Equation

We discuss the dispersionless Boussinesq type equation, which is equivalent to the Benney–Lax equation, being a system of equations of hydrodynamical type. This equation was discussed in [4]. The results include: A description of local and nonlocal Hamiltonian and symplectic structures, hierarchies of symmetries, hierarchies of conservation laws, recursion operators for symmetries and generating functions of conservation laws (cosymmetries). Highly interesting are the appearances of operators that send conservation laws and symmetries to each other but are neither Hamiltonian, nor symplectic. These operators give rise to a noncommutative infinite-dimensional algebra of recursion operators.

Vortex Line Representation for the Hydrodynamic Type Equations

In this paper we give a brief review of the recent results obtained by the author and his co-authors for description of three-dimensional vortical incompressible flows in the hydrodynamic type systems. For such flows we introduce a new mixed Lagrangian-Eulerian description - the so called vortex line representation (VLR), which corresponds to transfer to the curvilinear system of coordinates moving together with vortex lines. Introducing the VLR allows to establish the role of the Cauchy invariants from the point of view of the Hamiltonian description. In particular, these (Lagrangian) invariants, characterizing the property of frozenness of the generalized vorticity into fluids, are shown to represent the infinite (continuous) number of Casimirs for the so-called non-canonical Poisson brackets. The VLR allows to integrate partially the equations of motion, to exclude the infinite degeneracy due to frozenness of the primitive Poisson brackets and to establish in new variables the variational principle. It is shown that the original Euler equations for vortical flows coincides with the equations of motion of a charged compressible fluid moving due to a self-consistent electromagnetic field. Transition to the Lagrangian description in a new hydrodynamics is equivalent to the VLR. The VLR, as a mapping, turns out to be compressible that gives a new opportunity for collapse in fluid systems - breaking of vortex lines, resulting in infinite vorticity. It is shown that such process is possible for three-dimensional integrable hydrodynamics with the Hamiltonian where Ω is the vorticity. We also discuss some arguments in the favor of existence of such type of collapses for the Euler hydrodynamics, based on the results of some numerics.

Self-consistent turbulent convection in a magnetized plasma

It has been shown that low-frequency vortex convection, which is self-consistently developed in a magnetized plasma, may lead to nondiffusive transport processes similar to those observed in various systems of the plasma magnetic confinement. To theoretically analyze such a convection, an approach is proposed on the basis of the direct computer simulation of the quasi-two-dimensional dynamics of a weakly dissipative plasma with the use of adiabatically reduced hydrodynamic-type equations. The derived equations ensure the description of both relatively fast nonlinear convective flows and slower resulting transport processes and allow the simulation of the plasma evolution at sufficiently long times comparable with the plasma lifetime. The simulation shows that the development of the convection leads to the formation of nonlinear large-scale stochastic vortex structures, which exhibit broad power-law frequency and wavenumber spectra, as well as the non-Gaussian statistics of fluctuations, and corresponds to the notion of structure turbulence. The resulting transport processes are nonlocal and nondiffusive and have a number of characteristic properties similar to those observed in real experiments. The self-consistency of the pressure and density profiles in the plasma, L-H transitions, impurity pinch, etc., are among these properties.

Piecewise continuous distribution function method in the theory of wave disturbances of inhomogeneous gas

The system of hydrodynamic-type equations for a stratified gas in gravity field is derived from BGK equation by method of piecewise continuous distribution function. The obtained system of the equations generalizes the Navier–Stokes one at arbitrary Knudsen numbers. The problem of a wave disturbance propagation in a rarefied gas is explored. The verification of the model is made for a limiting case of a homogeneous medium. The phase velocity and attenuation coefficient values are in an agreement with former fluid mechanics theories; the attenuation behavior reproduces experiment and kinetics-based results at more wide range of the Knudsen numbers.

Whitham hierarchy in growth problems

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows distinguishing a class of exact solutions of the Laplacian growth problem in the multiply connected case. These solutions correspond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type, which are solvable by the generalized hodograph method.

Whitham hierarchy in growth problems

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows distinguishing a class of exact solutions of the Laplacian growth problem in the multiply connected case. These solutions correspond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type, which are solvable by the generalized hodograph method.

Whitham hierarchy in growth problems

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows distinguishing a class of exact solutions of the Laplacian growth problem in the multiply connected case. These solutions correspond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type, which are solvable by the generalized hodograph method.

Classifying Integrable Egoroff Hydrodynamic Chains

We introduce the notion of Egoroff hydrodynamic chains. We show how they are related to integrable (2+1)-dimensional equations of hydrodynamic type. We classify these equations in the simplest case. We find (2+1)-dimensional equations that are not just generalizations of the already known Khokhlov–Zabolotskaya and Boyer–Finley equations but are much more involved. These equations are parameterized by theta functions and by solutions of the Chazy equations. We obtain analogues of the dispersionless Hirota equations.

Extra dimensions and non‐linear equations

AbstractIt is shown how integrable non‐linear equations of hydrodynamic type, which arise as continuations of Dirac‐Born‐Infeld equations to the case where the target space is of lower dimension than the base space, are associated with linear diffusion equations in a space of double the number of space co‐ordinates.

The anomalous diffusion of wave disturbances in hydrodynamic-type systems

It is shown that solutions of many model non-linear equations of the hydrodynamic type (including the Navier-Stokes equations in the theory of a viscous fluid) in the form of localized wave disturbances undergo anomalous decay and the diffusion-type spreading under the action of Gaussian additive force noise and multiplicative parametric noise that depend randomly on time. The universality of this anomaly (i.e. its independence of the specific form of the equations and, in many respects, of the characteristic properties of the noise) is demonstrated; this reflects the generalized Galilean invariance of the hydrodynamic-type equations and the Gaussian property of random sources. The integrability of certain important hydrodynamic models enables the general conclusions to be supported by particular analytic results. Examples of expressions for the mean flow velocities, developed under the action of random forces on algebraic solitons of the completely integrable equations: equations for internal waves (the Benjamin-Ono equations), the two-dimensional Korteweg-de Vries equation (the Petviashvili-Kadomtseve equation), and the two-dimensional non-linear Schrödinger equation (the Davey-Stewartson equations), are presented in terms of the error function. It is established that for stochastic equations of the hydrodynamic type rapidly decaying multiplicative noise leads to normal Fick diffusion only when there is no additive force noise. Force noise produces anomalous diffusion of wave disturbances, which is described by Richardson's law (with a cubic relation between the temporal and squared spatial scales). The result obtained confirms the universal nature of Richardson's law, which has been demonstrated previously for many examples of turbulent and wave stochastic processes [1, 2].

ON THE INTEGRABILITY OF A CLASS OF MONGE–AMPÈRE EQUATIONS

We give the Lax representations for the elliptic, hyperbolic and homogeneous second order Monge–Ampère equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless Lax representation. A matrix dispersive Lax representation follows from the correspondence between sigma models, a two parameter equation for minimal surfaces and Monge–Ampère equations. Local as well nonlocal conserved densities are obtained.

Recursion operators of some equations of hydrodynamic type

We give a general method for constructing recursion operators for some equations of hydrodynamic type, admitting a nonstandard Lax representation. We give several examples for N=2 and N=3 containing the equations of shallow water waves and its generalizations with their first two general symmetries and their recursion operators. We also discuss a reduction of N+1 systems to N systems of some new equations of hydrodynamic type.