V. M. Shelkovich's research while affiliated with Saint-Petersburg State University of Architecture and Civil Engineering and other places
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Publications (75)
The theory of p-adic wavelets is presented. One-dimensional and multidimensional wavelet bases and their relation to the spectral theory of pseudodifferential operators are discussed. For the first time, bases of compactly supported eigenvectors for p-adic pseudodifferential operators were considered by V.S. Vladimirov. In contrast to real wavelets...
In our previous paper, the Haar multiresolution analysis (MRA) \(\{V_{j}\}_{j\in \mathbb {Z}}\) in \(L^{2}(\mathbb {A})\) was constructed, where \(\mathbb {A}\) is the adele ring. Since \(L^{2}(\mathbb {A})\) is the infinite tensor product of the spaces \(L^{2}({\mathbb {Q}}_{p})\), p=∞,2,3,…, the adelic MRA has some specific properties different f...
Recently there were presented new examples of MRAs based on p-adic and adelic wavelets. This is a note of the methodological nature. Its aim is embed all known examples MRAs into the measure-free scheme based on abstract Hilbert space formalism. On the one hand, such an embedding unifies the known examples of MRAs. On the other hand, the presented...
To solve nonlinear systems of conservation laws, we need a proper concept of weak solution. The aim of this paper is to explain how to derive integral identities for defining δ-shock type solutions in the sense of Schwartzian distributions. We consider two types of systems to compare our definitions. System (1.3) is a standard system admitting delt...
In this paper we continue to study {\it quasi associated homogeneous
distributions \rm{(}generalized functions\rm{)}} which were introduced in the
paper by V.M. Shelkovich, Associated and quasi associated homogeneous
distributions (generalized functions), J. Math. An. Appl., {\bf 338}, (2008),
48-70. [arXiv:math/0608669]. For the multidimensional c...
A study was conducted to construct adelic Haar multiresolution analysis (MRA) and wavelet bases. An efficient method for constructing wavelet bases, which is based on the notion of multiresolution analysis, was introduced in the real case. The study was conducted to demonstrate increasing applications using adele rings. It was essential to φ(p)(X p...
In the theory of -adic evolution pseudo-differential equations (with time variable and space variable ), we suggest a method of separation of variables (analogous to the classical Fourier method) which enables us to solve the Cauchy problems for a wide class of such equations. It reduces the solution of evolution pseudo-differential equations to th...
We construct an infinite-dimensional linear space [Formula: see text] of vector-valued distributions (generalized functions), or sequences, f*(x)=(f n (x)) finite from the left (i.e. f n (x)=0 for n<n 0 (f*)) whose components f n (x) belong to the linear span [Formula: see text] generated by the distributions δ (m-1) (x-c k ), P((x-c k ) -m ), x m-...
For one-dimensional system of zero-pressure gas dynamics, we derive the balance laws describing mass, momentum and energy transport from the area outside the δ-shock wave front onto its front. It is proved that the total mass and momentum of area outside δ-shock wave front and the the δ-shock wave front are conserved, while the kinetic energy of th...
This paper is devoted to wavelet analysis on the adele ring \({\mathbb{A}}\) and the theory of pseudo-differential operators. We develop the technique which gives the possibility to generalize finite-dimensional results of wavelet analysis to the case of adeles \({\mathbb{A}}\) by using infinite tensor products of Hilbert spaces. We prove that \(L^...
We study δ-shocks in a one-dimensional system of zero-pressure gas dynamics. In contrast to well-known papers (see References) this system is considered in the form of mass, momentum and energy conservation laws. In order to define such singular solutions, special integral identities are introduced which extend the concept of classical weak solutio...
We introduce integral identities to define delta-shock wave type solutions
for the multidimensional zero-pressure gas dynamics Using these integral
identities, the Rankine-Hugoniot conditions for delta-shocks are obtained. We
derive the balance laws describing mass, momentum, and energy transport from
the area outside the delta-shock wave front ont...
We study the propagation of δ-shock wave in a new type of system of conservation laws. The particular cases of this system are the system of nonlinear chromatography and the system for isotachophoresis.
We solve the Cauchy problems for p-adic linear and semi-linear evolutionary pseudo-differential equations (the time-variable t∈R and the space-variable ). Among the equations under consideration there are the heat type equation and the Schrödinger type equations (linear and nonlinear). To solve these problems, we develop the “variable separation me...
In this paper an infinite family of new compactly supported non-Haar p-adic wavelet bases in $
\mathcal{L}^2 (\mathbb{Q}_p^n )
$
\mathcal{L}^2 (\mathbb{Q}_p^n )
is constructed. We also study the connections between wavelet analysis and spectral analysis of p-adic pseudo-differential operators. A criterion for a multidimensional p-adic wavelet to be...
We describe all MRA-based p-adic compactly supported wavelet systems forming an orthogonal basis for L
2(ℚ
p
).
The notion of p-adic multiresolution analysis (MRA) is introduced. We discuss a “natural” refinement equation whose solution (a refinable function) is the characteristic function
of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of p characteristic functions of mutually disjoint discs of ra...
The algebraic aspects of δ-shock wave type solutions to multidimensional systems of conservation laws are studied. We show that any singular solution of the Cauchy problem generates some algebraic relations between its distributional components (“right” singular superpositions of distributions). We prove that the nonlinear flux-functions for δ-shoc...
We study the asymptotical behavior of the $p$-adic singular Fourier integrals $$ J_{\pi_{\alpha},m;\phi}(t) =\bigl< f_{\pi_{\alpha};m}(x)\chi_p(xt), \phi(x)\bigr> =F\big[f_{\pi_{\alpha};m}\phi\big](t), \quad |t|_p \to \infty, \quad t\in \bQ_p, $$ where $f_{\pi_{\alpha};m}\in {\cD}'(\bQ_p)$ is a {\em quasi associated homogeneous} distribution (gener...
In the present paper an infinite family of new compactly supported non-Haar p-adic wavelet bases in is constructed. These bases cannot be constructed in the framework of any of known theories. We use the wavelet bases in the following applications: in the theory of p-adic pseudo-differential operators and equations. The connections between wavelet...
This is a survey of some results and problems connected with the theory
of generalized solutions of quasi-linear conservation law systems which
can admit delta-shaped singularities. They are the so-called
\delta-shock wave type solutions and the recently introduced
\delta^{(n)}-shock wave type solutions, n=1,2,\dots, which cannot be
included in the...
New definitions of δ-shock-wave-type solutions are introduced for two (one-dimensional) types of hyperbolic systems of conservation laws. The
corresponding Rankine-Hugoniot conditions for δ-shocks are derived and their geometrical interpretation is given. Balance laws connected with “area” mass and momentum transportation
for δ-shocks are derived.
{\it $\delta$-Shock wave type solutions} in the multidimensional system of conservation laws $$ \rho_t + \nabla\cdot(\rho F(U))=0, \qquad (\rho U)_t + \nabla\cdot(\rho N(U))=0, \quad x\in \bR^n, $$ are studied, where $F=(F_j)$ is a given vector field, $N=(N_{jk})$ is a given tensor field, $F_j, N_{kj}:\bR^n \to \bR$, $j,k=1,...,n$; $\rho(x,t)\in \b...
A compactly supported orthonormal non-Haar p-adic wavelet basis is constructed that extensively use pseudodifferential operators. In the first theorem, the function system is an orthornormal wavelet basis in ℒ2(ℚp). In the second theorem, the function system is an orthonormal wavelet basis ℒ2(ℚ np).
It is well known that there are “nonclassical” situations where, in contrast to Lax’s and Glimm’s results, the Cauchy problem
for a system of conservation laws does not possess a weak L∞-solution except for some particular initial data. To solve the Cauchy problem in this “nonclassical” situation, it is necessary
to introduce new singular solutions...
The main goal of this paper is the development of the MRA theory in L-2(Q(p)). We described a wide class of p-adic refinement equations generating p-adic multiresolution analyses. A method for the construction of p-adic orthogonal wavelet bases within the framework of the MRA theory is suggested. A realization of this method is illustrated by an ex...
Using the definitions of δ - and δ ′-shocks for some systems of conservation laws, the corresponding Rankine–Hugoniot conditions are derived. We also derive the balance laws describing area, volume, mass and momentum transportation between the area outside the wave front and the wave front. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
We propose an algebraic theory which can be used for solving both linear and non-linear singular problems of -adic analysis related to -adic distributions (generalized functions). We construct the -adic Colombeau-Egorov algebra of generalized functions, in which Vladimirov's pseudo-differential operator plays the role of differentiation. This algeb...
In this paper, the notion of {\em $p$-adic multiresolution analysis (MRA)} is introduced. We use a ``natural'' refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is the sum of $p$ characteristic functions of disj...
In this paper we study some problems related with the theory of multidimensional $p$-adic wavelets in connection with the theory of multidimensional $p$-adic pseudo-differential operators (in the $p$-adic Lizorkin space). We introduce a new class of $n$-dimensional $p$-adic compactly supported wavelets. In one-dimensional case this class includes t...
There are many applications of p -adic analysis in quantum mechanics, string theory, stochastics, the theory of dynamical systems, and cognitive sciences (3-7, 9, 13-15). The systematic use of p -adic analysis in applications began with pioneering works of V.S. Vladimirov and I.V. Volovich. In p -adic analysis, which is related to the mappingp → �...
The points to be considered are, first, integral identities for the definition of multidimensional -shocks to systems of conservation laws; second, the Rankine-Hugoniot conditions for -shocks; third, studying the trans- portation and concentration processes related to -shocks; and fourth, solving the Cauchy problems admitting -shocks. The eects of...
In this paper, using the vanishing viscosity method, we construct a solution of the Riemann problem for the system of conservation lawsut+(u2)x=0,vt+2(uv)x=0,wt+2(v2+uw)x=0 with the initial data(u(x,0),v(x,0),w(x,0))={(u−,v−,w−),x0,(u+,v+,w+),x>0. This problem admits δ-, δ′-shock wave type solutions, and vacuum states. δ′-Shock is a new type of sin...
In this paper analysis of the concept of {\it associated homogeneous distributions} (generalized functions) is given, and some problems related to these distributions are solved. It is proved that (in the one-dimensional case) there exist {\it only} {\it associated homogeneous distributions} of order $k=1$. Next, we introduce a definition of {\it q...
In this article the p-adic Lizorkin spaces of test functions and distributions are introduced. Multi-dimensional Vladimirov’s
and Taibleson’s fractional operators, and a class of p-adic pseudo-differential operators are studied on these spaces. Since
the p-adic Lizorkin spaces are invariant under these operators, they can play a key role in conside...
Generalizing the classical definition of a weak L<sup>infin</sup>-solution definitions of delta- and delta'-shocks for systems of conservation laws were introduced and the Cauchy problems admitting such singular solutions were solved. In this paper we discuss and substantiate the validity and naturalness of the above-mentioned definitions. As these...
A concept of a new type of singular solutions to hyperbolic systems of conservation laws is introduced. It is so-called - (n) -shock wave, where - (n) is n-th derivative of the delta function. We introduce a definition of - 0 -shock wave type solution for the system ut + ¡ f(u) ¢ x = 0; vt + ¡ f 0 (u)v ¢ x = 0; wt + ¡ f 00 (u)v 2 + f 0 (u)w ¢ x = 0...
A concept of a new type of singular solutions to systems of conservation laws is introduced. It is so-called δ(n)-shock wave, where δ(n) is nth derivative of the Dirac delta function (n=1,2,…). In this paper the case n=1 is studied in details. We introduce a definition of δ′-shock wave type solution for the system Within the framework of this defin...
The algebraic aspect of singular solutions ( δ -shocks) to systems of conservation laws is studied. Namely, we show that singular solution of the Cauchy problem generates algebraic relations between distributional components of a singular solution ("right" singular superpositions of distributions). These "right" singular superpositions of distribut...
The p‐adic Lizorkin type spaces of test functions and distributions are introduced and a class of pseudo‐differential operators on this spaces are constructed. The p‐adic Lizorkin spaces are invariant under the above‐mentioned pseudo‐differential operators. This class of pseudo‐differential operators contains the Taibleson fractional operators. Sol...
The notion of quasi-asymptotics at infinity and at zero adapted to the case of p-adic distributions (generalized functions) are introduced, and p-adic analogs of Tauberian theorems for distributions are proved (1). We show that some properties of Vladimirov's pseudo-dierentia l operator D are connected with a prototype of the Tauberian type theorem...
In this paper some multidimensional Tauberian theorems for the Lizorkin distributions (without restriction on the support) are proved. Tauberian theorems of this type are connected with the Riesz fractional operators.
In this paper the p -adic Lizorkin spaces of test functions and distributions are introduced, and multidimensional Vladimirov's and Taibleson's fractional operators are studied on these spaces. Since the p -adic Lizorkin spaces are invariant under the Vladimirov and Taibleson operators, they can play a key role in considerations related to fraction...
The notion of the quasi-asymptotic adapted to the case of p-adic distributions (generalized functions) is introduced. p-adic analogs of Tauberian theorems for distributions are proved. We show that some properties of Vladimirov’s pseudo-differential operator D α are connected with a prototype of a Tauberian type theorem (with respect to distributio...
We introduce a concept of associated homogeneous p-adic distributions (generalized functions) and provide a mathematical description of all associated homogeneous distributions and their Fourier transform. We prove that any associated homogeneous distribution of degree πα(x) is periodic in the variable α.
We construct some versions of the Colombeau theory. In particular, we construct the Colombeau algebra generated by harmonic (or polyharmonic) regularizations of distributions connected with a half-space and by analytic regularizations of distributions connected with an octant. Unlike the standard Colombeau's scheme, our theory has new generalized f...
For two classes of hyperbolic systems of conservation laws new definitions of a delta-shock wave type solution are introduced. These two definitions give natural generalizations of the classical definition of the weak solutions. It is relevant to the notion of delta-shocks. The weak asymptotics method developed by the authors is used to describe th...
We introduce a new definition of aδ-shock wave type solution for a class of systems of conservation laws in the one-dimensional case. The weak asymptotics method developed by the authors is used to construct formulas describing the propagation and interaction of δ-shock waves. The dynamics of merging two δ-shocks is described by explicit formulas c...
The p-adic Colombeau-Egorov algebra of generalized functions on ℚnp is constructed. For generalized functions the operations of multiplication, Fourier-transform, convolution, taking pointvalues are defined. The operations of (fractional) partial differentiation and (fractional) partial integration are introduced by the Vladimirov's pseudodifferent...
In this paper the δ-shock front problem is studied. For some classes of hyperbolic systems of conservation laws (in several space dimension, too) we introduce the definitions of a δ-shock wave type solution relevant to the front problem. The Rankine–Hugoniot conditions for δ-shocks are analyzed from both geometrical and physical points of view. δ-S...
The definition of a delta-shock wave type solution to hyperbolic systems of conservation laws is introduced. This solution is a new type of singular solutions. The Rankine-Hugoniot conditions for delta-shock are derived
no. 176 Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs-gemeinschaft getragenen Sonderforschungsbereiches 611 an der Univer-sität Bonn entstanden und als Manuskript vervielfältigt worden. Abstract. The problem of the propagation of δ-shock waves in the multidimen-sional case a non-conservative form of zero-pressure gas dynamics...
The problem of defining δ-shock wave type solutions of hyperbolic systems of conservation laws in connection with the constructing singular superpositions (products) of distributions is studied. We illustrate this problem by constructing δ-shock wave type solutions for two systems, One of them, ut + (f(u) - v)x = 0, vt + (g(u)) x = 0, is a generali...
Introduced as a concept, constructed and studied are uniform p-adic distributions, theory of which is used in mathematical physics, in particular, in theory of p-adic strings and p-adic quantum mechanics. Power and order lemmas, linear independence and periodically associated uniform distributions lemmas are proved. Theorem giving description of al...
We propose a new regularization method — the weak asymptotics method — for investigating propagation and interaction of delta-shock waves of a hyperbolic system of conservation laws.
An associative commutative algebra including homogeneous and adjoint homogeneous distributions has been constructed. The products of distributions are continuous linear functionals. The conditions of solvability of strictly hyperbolic systems of quasilinear equations of the first order in the Maslov algebras are obtained.
An associative commutative algebra of distributions that contains homogeneous and associated homogeneous distributions is
constructed. This algebra is used to analyze generalized solutions to strictly hyperbolic partial differential equations.
Possible types of singularities are studied and the necessary (analogues of Hugoniót conditions for shock...
Using the deflnitions of -- and -0-shocks for the systems of conservation laws (12), (13), (39), the Rankine{Hugoniot conditions for -- and -0-shocks are derived. We present a construction of solutions to the Cauchy problems admitting -- and -0-shocks. In particular, the Riemann problem admitting shocks, --shocks, -0-shocks, and vacuum states is co...
We construct --shock type solutions of the Cauchy
We construct a definition of the weak solution to KdV type equations with small dispersion admitting the zero dispersion limit for soliton-like solutions. Using this definition, we obtain a system of equations (the limit problem as the dispersion tends to zero) that describes the soliton dynamics.
In this paper, using the vanishing viscosity method, a solution of the Riemann problem for the system of conservation laws ut + ¡ u2 ¢ x = 0; vt + 2 ¡ uv ¢ x = 0; wt + 2 ¡ v2 + uw ¢ x = 0 with the initial data (u(x;0);v(x;0);w(x;0)) = ‰ (u¡;v¡;w¡); x < 0; (u+;v+;w+); x > 0; is constructed. This problem admits a -0-shock wave type solution, which is...
Citations
... The limit as ε → 0 of the family (δ ε H ε ) can then be defined to be the product of the δ-distribution in the context of the weak asymptotic method, and one obtains approximate solutions to (1.1) for which one concludes that they converge weakly to a measure containing a δ-distribution. This approach has been used in [14,16,65] among many others, and is similar to the vanishing viscosity and the vanishing pressure methods used for example in [29,35,45]. One may also note that the family (δ ε H ε ) can be considered as an element of a Colombeau algebra, such as defined in [11]. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/273707172102153-1442268283445_Q64/Georgii-Omelyanov.jpg)
... By adaptation of definitions from [26] to the case of the field Q p (instead of the real field), a notion of the p-adic quasi associated homogeneous distribution was introduced in [1], [2] by Definition 2.2. (In [1], [2] these new distributions were named as associated homogeneous distributions). ...
... These transformations known as Lie group of transformation are continuous transformations associated with independent and dependent variables governing the PDEs. Lie group of transformations also known as Lie symmetries provide much flexibility in treating nonlinear systems, for example, one can use them to reduce the governing system to an equivalent system of ordinary differential equations (ODEs) 1,2 or one can reduce the governing system to an equivalent system autonomous PDEs 3,4 or a much easier system of PDEs. 5,6 Once the governing system is reduced to an equivalent system, one can look for exact solutions for these systems. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/277540628385812-1443182250804_Q64/Evgeniy-Panov.jpg)
... In this calculus, the crucial role is played by the fractional differentiation operator D α (the Vladimirov operator). The pseudodifferential equations over p-adic fields have been studied in numerous publications [3,8,12,[17][18][19]22,24,25,27,32,[34][35][36][37]. But up to now, in almost all models, only linear and semilinear pseudo-differential equations have been considered (see also [2,4,9,10,28,41,42]). ...
... These wavelet bases were obtained by relaxing the basis condition in the definition of an MRA and form Riesz bases without any dual wavelet systems. For some related works on wavelets and frames on Q p , we refer to [11,20,22,23]. On the other hand, Lang [24,25,26] constructed several examples of compactly supported wavelets for the Cantor dyadic group. ...
Reference: Wavelet bi-frames on local fields
![[object Object]](https://i1.rgstatic.net/ii/profile.image/272732063531056-1442035799736_Q64/Sergei-Kozyrev.jpg)
... It is well known that there are "nonclassical" situations where, in contrast to Lax's and Glimm's classical results, the Cauchy problem for a system of conservation laws does not possess a weak L ∞solution or possesses it for some particular initial data. In order to solve the Cauchy problem in this "nonclassical" situation, it is necessary to introduce new singular solutions called δ-shocks (see [1], [8]- [10], [13]- [17], [22], [24]- [31], [39]- [41], [44]- [49] and the references therein), which is a solution such that its components contain Dirac measures. ...
... In order to solve this Cauchy problem in the framework of nonclassical solutions, it is necessary to construct the solution for this Cauchy problem with "strong singularities". Fortunately, in the past two decades, δ-shock wave was introduced to describe this phenomenon, see [3,6,10,14,26,30,31,35,38] for classical fluids and see [37,13] for relativistic Euler equations of Chaplygin gases. Roughly speaking, δ-shock wave is a kind of discontinuity, on which at least one of the state variables may develop an extreme concentration in the form of a weighted Dirac delta function with the discontinuity as its support. ...
... The theory of minimally supported frequency wavelets for LCAG and local fields has been developed by Benedetto and Benedetto [5] while the peculiar examples of wavelets for local fields have been provided by Benedetto in [4]. Next for the other local field of characteristic 0, namely, p-adic field Q p , a rigorous study of wavelet theory on Q p has been done by Albeverio, Kozyrev, Skopina, Khrennikov, van der Walt, Shelkovich et al. in a series of papers [1,2,[20][21][22][23][24]. Further, regarding the wavelet functions corresponding to the quincunx Haar MRA, Albeverio and Skopina in [2] provided an explicit description and found a connection between quincunx Haar bases and two-dimensional separable Haar MRA while Packer in [29] discussed about the projective multiresolution analyses and a projective MRA corresponding to the quincunx lattice. ...
... For the review of p-adic wavelets see [172]. Important contributions in p-adic wavelet theory were done also by Albeverio, Benedetto, Khrennikov, Shelkovich, Skopina, Evdokimov [41,42,6,7,143,144,222,145,15]. Wavelets on locally compact abelian groups (in particular Kantor dyadic group) were considered in [127,128,113], and wavelets on adeles were investigated in [187,188,151]. ...
... In this section we fix the notation and collect some basic results on p-adic analysis that we will use through the article. For a detailed exposition on p-adic analysis the reader may consult [1], [30], [33]. For a quick review of p-adic analysis the reader may consult [5], [24]. ...
