We study the initial-boundary value problem for the one-dimensional Oskolkov pseudoparabolic equation of viscoelasticity with a nonlinear convective term and a linear absorption term. The absorption term depends on a …
We study the initial-boundary value problem for the one-dimensional Oskolkov pseudoparabolic equation of viscoelasticity with a nonlinear convective term and a linear absorption term. The absorption term depends on a positive integer parameter n and, as n → + ∞ , converges weakly * to the expression incorporating the Dirac deltafunction, which models an instant absorption at the initial moment of time. We prove that the infinitesimal initial layer, associated with the Dirac delta function, is formed as n → + ∞ , and that the family of regular weak solutions of the original problem converges to the strong solution of a two-scale microscopic-macroscopic model. The main novelty of the article consists of taking into account of the effect of convection. In the final section, some possible generalizations and applications are briefly discussed, in particular with regard to active fluids.
In the present article, we study singular limits of weak energy solutions to non-instantaneous single-and multi-impulsive advection-diffusion-reaction equation as impulsive source terms collapse to time-dependent the Dirac delta-functions, i.e., to …
In the present article, we study singular limits of weak energy solutions to non-instantaneous single-and multi-impulsive advection-diffusion-reaction equation as impulsive source terms collapse to time-dependent the Dirac delta-functions, i.e., to instant impulses. We establish that the limiting functions are the solutions of the instantaneous impulsive advection-diffusion-reaction equations. Besides, in the multi-impulsive case we find the effect of transition to an equilibrium as a number of impulses grows infinitely. The results can be applied to further modeling of such processes as frost-quakes in glaciology.
Existence, uniqueness and stability of kinetic and entropy solutions to the boundary value problem associated with the Kolmogorov-type, genuinely nonlinear, degenerate hyperbolic–parabolic (ultra-parabolic) equation with a smooth source term is …
Existence, uniqueness and stability of kinetic and entropy solutions to the boundary value problem associated with the Kolmogorov-type, genuinely nonlinear, degenerate hyperbolic–parabolic (ultra-parabolic) equation with a smooth source term is established. In addition, we consider the case when the source term contains a small positive parameter and collapses to the Dirac delta-function, as this parameter tends to zero. In this case, the limiting passage from the original equation with the smooth source to the impulsive ultra-parabolic equation is investigated and the formal limit is rigorously justified. Our proofs rely on the use of kinetic equations and the compensated compactness method for genuinely nonlinear balance laws.
We study the Cauchy-Dirichlet problem for the p(x)-Laplacian equation with a regular finite nonlinear minor term.The minor term depends on a small parameter ε > 0 and, as ε → …
We study the Cauchy-Dirichlet problem for the p(x)-Laplacian equation with a regular finite nonlinear minor term.The minor term depends on a small parameter ε > 0 and, as ε → 0, converges weakly to the expression incorporating the Dirac delta function, which models a shock (impulsive) loading.We establish that the shock layer, associated with the Dirac delta function, is formed as ε → 0, and that the family of weak solutions of the original problem converges to a solution of a two-scale microscopicmacroscopic model.This model consists of two equations and the set of initial and boundary conditions, so that the 'outer' macroscopic solution beyond the shock layer is governed by the usual homogeneous p(x)-Laplacian equation, while the shock layer solution is defined on the microscopic level and obeys the ordinary differential equation derived from the microstructure of the shock layer profile.
Abstract The Cauchy-Dirichlet problem for the genuinely nonlinear ultra-parabolic equation with the piece-wise smooth minor term is considered. The minor term depends on a small positive parameter and collapses to …
Abstract The Cauchy-Dirichlet problem for the genuinely nonlinear ultra-parabolic equation with the piece-wise smooth minor term is considered. The minor term depends on a small positive parameter and collapses to the one-sided Dirac delta function as this parameter tends to zero. As the result, we arrive at the limiting initial-boundary value problem for the impulsive ultra-parabolic equation. The peculiarity is that the standard entropy solution of the problem for the impulsive equation generally is not unique. In this report, we propose a rule for selecting the ‘proper’ entropy solution, relying on the limiting procedure in the original problem incorporating the smooth minor term.
Existence, uniqueness and stability of kinetic and entropy solutions to the boundary value problem for the Kolmogorov-type genuinely nonlinear ultra-parabolic equation with a smooth source term is established. After this, …
Existence, uniqueness and stability of kinetic and entropy solutions to the boundary value problem for the Kolmogorov-type genuinely nonlinear ultra-parabolic equation with a smooth source term is established. After this, we consider the case when the source term contains a small positive parameter and collapses to the Dirac delta-function, as this parameter tends to zero. In this case, the limiting passage from the original equation with the smooth source to the impulsive ultra-parabolic equation is fulfilled and rigorously justified. The proofs rely on the method of kinetic equation and on the compensated compactness techniques for genuinely nonlinear equations.
In the present paper, we derive the kinetic equation and impulsive condition and formulate a class of kinetic solutions to impulsive ultra-parabolic equations. Here ultra-parabolic equations are linked with diffusion …
In the present paper, we derive the kinetic equation and impulsive condition and formulate a class of kinetic solutions to impulsive ultra-parabolic equations. Here ultra-parabolic equations are linked with diffusion processes with inertia (convective heat transfer) on a shock wave front.
In this paper, anisotropic Sobolev — Slobodetskii spaces in poly-cylindrical domains of any dimension N are considered. In the first part of the paper we revisit the well-known Lions — …
In this paper, anisotropic Sobolev — Slobodetskii spaces in poly-cylindrical domains of any dimension N are considered. In the first part of the paper we revisit the well-known Lions — Magenes Trace Theorem (1961) and, naturally, extend regularity results for the trace and lift operators onto the anisotropic case. As a byproduct, we build a generalization of the Kruzhkov — Korolev Trace Theorem for the first-order Sobolev Spaces (1985). In the second part of the paper we observe the nonhomogeneous Dirichlet, Neumann, and Robin problems for p-elliptic equations. The well-posedness theory for these problems can be successfully constructed using isotropic theory, and the corresponding results are outlined in the paper. Clearly, in such a unilateral approach, the anisotropic features are ignored and the results are far beyond the critical regularity. In the paper, the refinement of the trace theorem is done by the constructed extension.DOI 10.14258/izvasu(2018)4-19
In the present paper we have proved the existence of quasi-solutions of genuinely nonlinear forward-backward ultra-parabolic equations. Quasi-solutions are obtained with the help of the vanishing anisotropic temporal diffusion method. …
In the present paper we have proved the existence of quasi-solutions of genuinely nonlinear forward-backward ultra-parabolic equations. Quasi-solutions are obtained with the help of the vanishing anisotropic temporal diffusion method. Moreover, at the present stage of our research we assume that various choices of temporal artificial diffusion coefficients lead to entropy solutions or to quasi-solutions. The latter assumption is the subject of our further scientific research.
In order to study weak limits of quadratic expressions of oscillatory solutions of partial differential equations, there was proposed a construction of H -measures defined on the space of positions …
In order to study weak limits of quadratic expressions of oscillatory solutions of partial differential equations, there was proposed a construction of H -measures defined on the space of positions and frequencies. The present paper is devoted to the investigation of the Tartar equation which describes the evolution of the H -measure μ t associated with a sequence of oscillatory solutions of the linear transport equation in cases when a given solenoidal velocity field v ( x , t ) is sufficiently smooth. Here, ( t, x , y ) ∈ (0, T ) × Ω × S 1 , 0 < T < +∞, Ω is a bounded open subset of R 2 and S 1 is the unit circle in R 2 , given coefficients Y ij = Y ij ( y ) are infinitely smooth. Assuming that v belongs to , we establish the well posedness of Cauchy problem for the Tartar equation in the same measure class as the H -measures are in. For this purpose, we develop and use an extension of the theory of Lagrange coordinates for a case of non-smooth solenoidal velocity fields.
Introductory material Auxiliary propositions Linear equations with discontinuous coefficients Linear equations with smooth coefficients Quasi-linear equations with principal part in divergence form Quasi-linear equations of general form Systems of linear …
Introductory material Auxiliary propositions Linear equations with discontinuous coefficients Linear equations with smooth coefficients Quasi-linear equations with principal part in divergence form Quasi-linear equations of general form Systems of linear and quasi-linear equations Bibliography.
We consider scalar conservation laws with nonlinear singular sources with a concentration effect at the origin.We assume that the flux A is not degenerated and we study whether it is …
We consider scalar conservation laws with nonlinear singular sources with a concentration effect at the origin.We assume that the flux A is not degenerated and we study whether it is possible to define a well-posed limit problem.We prove that when A is strictly monotonic then the limit problem is well-defined and has a unique solution.The definition of this limit problem involves a layer which is shown to be very stable.But when A is not monotonic this problem can be unstable.Indeed we can construct two sequences of approximate solutions which converge to two different functions although their initial values coincide in the limit.
Abstract This book gives a general presentation of the mathematical and numerical connections kinetic theory and conservation laws based on several earlier works with P. L. Lions and E. Tadmor, …
Abstract This book gives a general presentation of the mathematical and numerical connections kinetic theory and conservation laws based on several earlier works with P. L. Lions and E. Tadmor, as well as on more recent developments. The kinetic formalism approach allows the reader to consider Partial Differential Equations, such as some nonlinear conservation laws, as linear kinetic (or semi-kinetic) equations acting on a nonlinear quantity. It also aids the reader with using Fourier transform, regularisation, and moments methods to provide new approaches for proving uniqueness, regularizing effects, and a priori bounds. Special care has been given to introduce basic tools, including the classical Boltzmann formalism to derive compressible fluid dynamics, the study of oscillatons through the kinetic defect measure, and an elementary construction of solutions to scalar conservation laws. More advanced material contains regularizing effects through averaging lemmas, existence of global large solutions to isentropic gas dynamics, and a new uniqueness proof for scalar conservation laws. Sections are also devoted to the derivation of numerical approaches, the 'kinetic schemes', and the analysis of their theoretical properties.
In this note we introduce a new class of abstract impulsive differential equations for which the impulses are not instantaneous. We introduce the concepts of mild and classical solution and …
In this note we introduce a new class of abstract impulsive differential equations for which the impulses are not instantaneous. We introduce the concepts of mild and classical solution and we establish some results on the existence of these types of solutions. An example involving a partial differential equation is presented.
We study the Cauchy-Dirichlet problem for the p(x)-Laplacian equation with a regular finite nonlinear minor term.The minor term depends on a small parameter ε > 0 and, as ε → …
We study the Cauchy-Dirichlet problem for the p(x)-Laplacian equation with a regular finite nonlinear minor term.The minor term depends on a small parameter ε > 0 and, as ε → 0, converges weakly to the expression incorporating the Dirac delta function, which models a shock (impulsive) loading.We establish that the shock layer, associated with the Dirac delta function, is formed as ε → 0, and that the family of weak solutions of the original problem converges to a solution of a two-scale microscopicmacroscopic model.This model consists of two equations and the set of initial and boundary conditions, so that the 'outer' macroscopic solution beyond the shock layer is governed by the usual homogeneous p(x)-Laplacian equation, while the shock layer solution is defined on the microscopic level and obeys the ordinary differential equation derived from the microstructure of the shock layer profile.
There is a situation such that, when a function ƒ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">) is combined with the Dirac delta function δ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), the usual formula <img src="/img/revistas/rbef/v31n4/a04form01.gif" align="absmiddle">does not hold. …
There is a situation such that, when a function ƒ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">) is combined with the Dirac delta function δ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), the usual formula <img src="/img/revistas/rbef/v31n4/a04form01.gif" align="absmiddle">does not hold. A similar situation may also be encountered with the derivative of the delta function δ'(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), regarding the validity of <img src="/img/revistas/rbef/v31n4/a04form02.gif" align="absmiddle">. We present an overview of such unusual situations and elucidate their underlying mechanisms. We discuss implications of the situations regarding the transmission-reflection problem of one-dimensional quantum mechanics.
General description of impulsive differential systems linear systems stability of solutions periodic and almost periodic impulsive systems integral sets of impulsive systems optimal control in impulsive systems asymptotic study of …
General description of impulsive differential systems linear systems stability of solutions periodic and almost periodic impulsive systems integral sets of impulsive systems optimal control in impulsive systems asymptotic study of oscillations in impulsive systems a periodic and almost periodic impulsive system.
This article is devoted to nonlinear ordinary differential equations with additive or multiplicative terms consisting of Dirac delta functions or derivatives thereof. Regularizing the delta function terms produces a family …
This article is devoted to nonlinear ordinary differential equations with additive or multiplicative terms consisting of Dirac delta functions or derivatives thereof. Regularizing the delta function terms produces a family of smooth solutions. Conditions on the nonlinear terms, relating to the order of the derivatives of the delta function part, are established so that the regularized solutions converge to a limiting distribution.
In this note we discuss the usage of the Dirac $\delta$ function in models of phase oscillators with pulsatile inputs. Many authors use a product of the delta function and …
In this note we discuss the usage of the Dirac $\delta$ function in models of phase oscillators with pulsatile inputs. Many authors use a product of the delta function and the phase response curve in the right hand side of an ODE to describe the discontinuous phase dynamics in such systems. We point out that this notation has to be treated with care as it is ambiguous. We argue that the presumably most canonical interpretation does not lead to the intended behaviour in many cases.
We consider a model for non-static groundwater flow where the saturation-pressure relation is extended by a dynamic term. This approach, together with a convective term due to gravity, results in …
We consider a model for non-static groundwater flow where the saturation-pressure relation is extended by a dynamic term. This approach, together with a convective term due to gravity, results in a pseudo-parabolic Burgers type equation. We give a rigorous study of global travelling-wave solutions, with emphasis on the role played by the dynamic term and the appearance of fronts.
The monograph is devoted to the study of initial-boundary-value problems for multi-dimensional Sobolev-type equations over bounded domains. The authors consider both specific initial-boundary-value problems and abstract Cauchy problems for first-order …
The monograph is devoted to the study of initial-boundary-value problems for multi-dimensional Sobolev-type equations over bounded domains. The authors consider both specific initial-boundary-value problems and abstract Cauchy problems for first-order (in the time variable) differential equations with nonlinear operator coefficients with respect to spatial variables. The main aim of the monograph is to obtain sufficient conditions for global (in time) solvability, to obtain sufficient conditions for blow-up of solutions at finite time, and to derive upper and lower estimates for the blow-up time. The abstract results apply to a large variety of problems. Thus, the well-known Benjamin-Bona-Mahony-Burgers equation and Rosenau-Burgers equations with sources and many other physical problems are considered as examples. Moreover, the method proposed for studying blow-up phenomena for nonlinear Sobolev-type equations is applied to equations which play an important role in physics. For instance, several examples describe different electrical breakdown mechanisms in crystal semiconductors, as well as the breakdown in the presence of sources of free charges in a self-consistent electric field. The monograph contains a vast list of references (440 items) and gives an overall view of the contemporary state-of-the-art of the mathematical modeling of various important problems arising in physics. Since the list of references contains many papers which have been published previously only in Russian research journals, it may also serve as a guide to the Russian literature.
In the present paper we deal with kinetic and entropy solutions of quasilinear impulsive hyperbolic equations. The genuine nonlinearity condition enables to prove the existence of one-sided traces of these …
In the present paper we deal with kinetic and entropy solutions of quasilinear impulsive hyperbolic equations. The genuine nonlinearity condition enables to prove the existence of one-sided traces of these solutions on fixed-time hyperplanes. The latter fact provides the impulsive condition. This type of equations can be used in the fluctuating hydrodynamics.
In the paper, we consider the existence of the solution of the second-order impulsive differential equations with inconstant coefficients. We change the second-order impulsive partial differential equation into the equivalent …
In the paper, we consider the existence of the solution of the second-order impulsive differential equations with inconstant coefficients. We change the second-order impulsive partial differential equation into the equivalent equation by transformation. By using the critical point theory of variational method and Lax-Milgram theorem, we obtain new results for the existence of the solution of the impulsive partial differential equations.
In this paper we construct a theory of generalized solutions in the large of Cauchy's problem for the equations
In this paper we construct a theory of generalized solutions in the large of Cauchy's problem for the equations
Many basic equations in physics take the following form:Ot -(u) O, (1) where x E Rd, t > O, u-(Ul,...,us) represents the local density of the investigated quantities and f= …
Many basic equations in physics take the following form:Ot -(u) O, (1) where x E Rd, t > O, u-(Ul,...,us) represents the local density of the investigated quantities and f= (fl,"',fd) is the flux vector.Such an equation is often used to describe the conservation of the density u in the evolution of physical process.Common examples are conservation of mass, balance of momentum, and balance of energy.Thus, (1) is called a conservation law (equation).In the case s 1, (1) becomes a first order hyperbolic PDE, which we may call a scalar conservation law (equation).After certain preparation, the authors consider the Cauchy problem for the scalar conservation law:---+ V .f(u) 0,in R + xNd, (2) u(0,.u0, in Rd, with f-(fi,'",fd), fj ca(R) and Uo:Rd.It is well-known that, in general, (2) does not admit a classical solution even if both f and u 0 are smooth.Thus, one seeks weak solutions to (2), by which we mean a function u Loc(R + d) satisfying 0 Rd for all p C10(R Rd).To obtain a weak solution of (2), one natural method is the so-called vanishing of viscosity.It introduces the parabolic perturbation to equation (2), which amounts to the introduction of the viscosity.Thus, one considers the
We prove that the sequence of averaged quantities R R m un(x, p) ρ(p)dp, is strongly precompact in L 2 loc (R d ), where ρ ∈ L 2 c …
We prove that the sequence of averaged quantities R R m un(x, p) ρ(p)dp, is strongly precompact in L 2 loc (R d ), where ρ ∈ L 2 c (R m ), and un ∈ L 2 (R m ; L s (R d )), s ≥ 2, are weak solutions to differential operator equations with variable coefficients.In particular, this includes differential operators of hyperbolic, parabolic or ultraparabolic type, but also fractional differential operators.If s > 2 then the coefficients can be discontinuous with respect to the space variable x ∈ R d , otherwise, the coefficients are continuous functions.In order to obtain the result we prove a representation theorem for an extension of the H-measures.Content 1. Introduction 239 2. Statement of the main result 243 3. Auxiliary results 246 4. Proof of the main theorem 253 5. Ultra-parabolic equation with discontinuous coefficients 255 References 258
In this article, a solution of a nonlinear pseudoparabolic equation is constructed as a singular limit of a sequence of solutions of quasilinear hyperbolic equations. If a system with cross …
In this article, a solution of a nonlinear pseudoparabolic equation is constructed as a singular limit of a sequence of solutions of quasilinear hyperbolic equations. If a system with cross diffusion, modelling the reaction and diffusion of two biological, chemical, or physical substances, is reduced then such an hyperbolic equation is obtained. For regular solutions even uniqueness can be shown, although the needed regularity can only be proved in two dimensions.
We prove that if traceability conditions are fulfilled then a weak solution h ∈ L ∞ (ℝ + × ℝ d × ℝ) to the ultra-parabolic transport equation [Formula: see …
We prove that if traceability conditions are fulfilled then a weak solution h ∈ L ∞ (ℝ + × ℝ d × ℝ) to the ultra-parabolic transport equation [Formula: see text] is such that for every [Formula: see text], the velocity averaged quantity ∫ ℝ h(t, x, λ) ρ(λ)dλ admits the strong [Formula: see text]-limit as t → 0, i.e. there exist [Formula: see text] and set E ⊂ ℝ + of full measure such that for every [Formula: see text], [Formula: see text] As a corollary, under the traceability conditions, we prove the existence of strong traces for entropy solutions to ultra-parabolic equations in heterogeneous media.
We solve the initial and boundary condition problem for a general first order quasilinear equation in several space variables by using a vanishing viscosity method and give a definition which …
We solve the initial and boundary condition problem for a general first order quasilinear equation in several space variables by using a vanishing viscosity method and give a definition which characterizes the obtained solution.
Sequences of measure-valued solutions of a non-degenerate quasilinear equation of the first order are shown to be strongly precompact in the general case, when the flow functions contain independent variables …
Sequences of measure-valued solutions of a non-degenerate quasilinear equation of the first order are shown to be strongly precompact in the general case, when the flow functions contain independent variables and are merely continuous.