Search

Your search keyword '"Vitali Vougalter"' showing total 85 results
Start Over Author "Vitali Vougalter"
85 results on '"Vitali Vougalter"'

1. Linear and nonlinear non-Fredholm operators and their applications

Author

Messoud Efendiev and Vitali Vougalter

Subjects
solvability conditions, non-fredholm operators, anomalous diffusion, nonnegativity of solutions, Mathematics, QA1-939, Applied mathematics. Quantitative methods, T57-57.97
Abstract

In this survey we discuss the recent results on the existence in the sense of sequences of solutions for certain elliptic problems containing the non-Fredholm operators. First of all, we deal with the solvability in the sense of sequences for some fourth order non-Fredholm operators, such that the methods of the spectral and scattering theory for Schrödinger type operators are used for the analysis. Moreover, we present the easily verifiable necessary condition of the preservation of the nonnegativity of the solutions of a system of parabolic equations in the case of the anomalous diffusion with the negative Laplacian in a fractional power in one dimension, which imposes the necessary form of such system of equations that must be studied mathematically. This class of systems of PDEs has a wide range of applications. We conclude the survey with several new results nowhere published concerning the solvability in the sense of sequences for the generalized Poisson type equation with a scalar potential.

Published
2022
Full Text
View/download PDF

2. A model of miscible liquids in porous media

Author

Karam Allali, Vitaly Volpert, and Vitali Vougalter

Subjects
Darcy approximation, Korteweg stress, miscible liquids, porous media, Mathematics, QA1-939
Abstract

In this article we study the interaction of two miscible liquids in porous media. The model consists of hydrodynamic equations with the Korteweg stress terms coupled with the reaction-diffusion equation for the concentration. We assume that the fluid is incompressible and its motion is described by the Darcy law. The global existence and uniqueness of solutions is established for some optimal conditions on the reaction source term and external force functions. Numerical simulations are performed to show the behavior of two miscible liquids subjected to the Korteweg stress.

Published
2015

3. Solvability in the sense of sequences to some non-Fredholm operators

Author

Vitaly Volpert and Vitali Vougalter

Subjects
Solvability conditions, non Fredholm operators, Sobolev spaces, Mathematics, QA1-939
Abstract

We study the solvability of certain linear nonhomogeneous elliptic problems and show that under reasonable technical conditions the convergence in $L^2(mathbb{R}^d)$ of their right sides implies the existence and the convergence in $H^2(mathbb{R}^d)$ of the solutions. The equations involve second order differential operators without Fredholm property and we use the methods of spectral and scattering theory for Schrodinger type operators analogously to our preceding work [17].

Published
2013

11. Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion

Author

Messoud Efendiev and Vitali Vougalter

Subjects
Anomalous diffusion, Differential equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Type (model theory), Fixed point, Differential operator, 01 natural sciences, Integro-differential Equations, Mixed Diffusion, Non Fredholm Operators, Solvability Conditions, 010101 applied mathematics, Elliptic curve, Convergence (routing), 0101 mathematics, Diffusion (business), Analysis, Mathematics
Abstract

We establish the existence in the sense of sequences of solutions for certain integro-differential type equations in two dimensions involving the normal diffusion in one direction and the anomalous diffusion in the other direction in H 2 ( R 2 ) via the fixed point technique. The elliptic equation contains a second order differential operator without the Fredholm property. It is proved that, under the reasonable technical conditions, the convergence in L 1 ( R 2 ) of the integral kernels implies the existence and convergence in H 2 ( R 2 ) of the solutions.

Published
2021

12. Method of Monotone Solutions for Reaction-Diffusion Equations

Author

Vitaly Volpert, Vitali Vougalter, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Multi-scale modelling of cell dynamics : application to hematopoiesis (DRACULA), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), Peoples Friendship University of Russia [RUDN University] (RUDN), University of Toronto, Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Camille Jordan (ICJ), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL)

Subjects
Statistics and Probability, Degree (graph theory), Function space, Applied Mathematics, General Mathematics, AMS subject classification: 35K57, 010102 general mathematics, 35J61, System of linear equations, 01 natural sciences, 010305 fluids & plasmas, Elliptic operator, Monotone polygon, 0103 physical sciences, Reaction–diffusion system, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], A priori and a posteriori, Applied mathematics, 47H11, 0101 mathematics, Mathematics
Abstract

International audience; Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray-Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reaction-diffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and nonmonotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given.

Published
2021

13. Solvability in the sense of sequences for some fourth order non-Fredholm operators

Author

Vitali Vougalter and Messoud Efendiev

Subjects
Pure mathematics, Applied Mathematics, 010102 general mathematics, Type (model theory), Differential operator, 01 natural sciences, Non-fredholm Operators, Sobolev Spaces, Solvability Conditions, 010101 applied mathematics, symbols.namesake, Dimension (vector space), Convergence (routing), symbols, Order (group theory), Scattering theory, 0101 mathematics, Constant (mathematics), Analysis, Schrödinger's cat, Mathematics
Abstract

We study solvability of some linear nonhomogeneous elliptic problems and establish that under reasonable technical conditions the convergence in L 2 ( R d ) of their right sides implies the existence and the convergence in H 4 ( R d ) of the solutions. The problems contain the squares of the sums of second order non-Fredholm differential operators and we use the methods of the spectral and scattering theory for Schrodinger type operators. We especially emphasize that here we deal with the fourth order operators in contrast to the second order operators in [29] and investigate the dependence of the solvability conditions on the dimension of our problem when the constant a = 0 . We also consider the case of solvability with a single potential in an arbitrary dimension.

Published
2021

14. Solvability of some integro-differential equations with concentrated sources

Author

Vitali Vougalter

Subjects
Numerical Analysis, Work (thermodynamics), Differential equation, Quantitative Biology::Tissues and Organs, Applied Mathematics, 010102 general mathematics, Dirac delta function, 01 natural sciences, Quantitative Biology::Cell Behavior, Term (time), 010101 applied mathematics, Sobolev space, Computational Mathematics, symbols.namesake, Normal diffusion, symbols, 0101 mathematics, Analysis, Mathematics, Mathematical physics
Abstract

The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function. The proof...

Published
2020

16. On Solvability in the Sense of Sequences for some Non-Fredholm Operators with Drift and Anomalous Diffusion

Author

Vitali Vougalter

Subjects
Statistics and Probability, Anomalous diffusion, Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, Differential operator, 01 natural sciences, Fractional power, 010305 fluids & plasmas, Sobolev space, Regularization (physics), 0103 physical sciences, Periodic boundary conditions, 0101 mathematics, Laplace operator, Real line, Mathematics
Abstract

We study the solvability of certain linear nonhomogeneous elliptic equations and establish that, under some technical assumptions, the L2-convergence of the right-hand sides yields the existence and convergence of solutions in an appropriate Sobolev space. The problems involve differential operators with or without Fredholm property, in particular, the one-dimensional negative Laplacian in a fractional power, on the whole real line or on a finite interval with periodic boundary conditions. We prove that the presence of the transport term in these equations provides regularization of the solutions.

Published
2020

17. Solvability of some integro-differential equations with anomalous diffusion in higher dimensions

Author

Vitali Vougalter

Subjects
Numerical Analysis, Differential equation, Anomalous diffusion, Applied Mathematics, 010102 general mathematics, Mathematical analysis, 01 natural sciences, Fractional power, 010101 applied mathematics, Sobolev space, Computational Mathematics, 0101 mathematics, Laplace operator, Analysis, Mathematics
Abstract

The article deals with the studies of the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power ...

Published
2020

24. Solvability of some integro-differential equations with anomalous diffusion and transport

Author

Vitaly Volpert, Vitali Vougalter, Department of Mathematics [University of Toronto], University of Toronto, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Multi-scale modelling of cell dynamics : application to hematopoiesis (DRACULA), Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Peoples Friendship University of Russia [RUDN University] (RUDN), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Modélisation multi-échelle des dynamiques cellulaires : application à l'hématopoïese (DRACULA), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-Inria Lyon, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Integro-differential equations, Algebra and Number Theory, Differential equation, Anomalous diffusion, 010102 general mathematics, Mathematical analysis, 35R09, Sobolev spaces Mathematics Subject Classification 35R11, Fixed point, 01 natural sciences, Fractional power, Term (time), Sobolev space, Elliptic operator, Non Fredholm operators, 35K57, 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 010306 general physics, Laplace operator, Mathematical Physics, Analysis, Mathematics
Abstract

International audience; The article deals with the existence of solutions of an integro-differential equation in the case of anomalous diffusion with the negative Laplace operator in a fractional power in the presence of the transport term. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without the Fredholm property in unbounded domains are used. We discuss how the introduction of the transport term impacts the regularity of solutions.

Published
2021

25. On the Necessary Conditions for Preserving the Nonnegative Cone: Mixed Diffusion

Author

Messoud Efendiev and Vitali Vougalter

Subjects
Cone (topology), Dimension (vector space), Applied mathematics, Verifiable secret sharing, Diffusion (business), Space (mathematics), System of linear equations, Laplace operator, Parabolic partial differential equation, Mathematics
Abstract

The chapter deals with the easily verifiable necessary condition of the preservation of the nonnegativity of the solutions of a system of parabolic equations in the case of the mixed diffusion when the standard Laplacian in the first m variables is added to the Laplace operator in the rest of the variables in a fractional power in the space of an arbitrary dimension. This necessary condition is crucial for the applied analysis community since it imposes the necessary form of the system of equations that must be treated mathematically.

Published
2021

26. Solvability in the Sense of Sequences for Some Non-Fredholm Operators in Higher Dimensions

Author

Vitali Vougalter and Vitaly Volpert

Subjects
Sobolev space, symbols.namesake, Pure mathematics, Square root, Convergence (routing), symbols, Order (ring theory), Scattering theory, Type (model theory), Differential operator, Schrödinger's cat, Mathematics
Abstract

We study solvability of certain linear nonhomogeneous elliptic problems and establish that under reasonable technical assumptions the convergence in \(L^{2}({\mathbb R}^{d})\) of their right sides implies the existence and the convergence in \(H^{1}({\mathbb R}^{d})\) of the solutions. The equations involve the square roots of the sums of second order non-Fredholm differential operators and we rely on the methods of the spectral and scattering theory for Schrodinger type operators similarly to our earlier work [26].

Published
2020

27. Existence of solutions for some non‐Fredholm integro‐differential equations with the bi‐Laplacian

Author

Vitali Vougalter, Vitaly Volpert, Department of Mathematics [University of Toronto], University of Toronto, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Multi-scale modelling of cell dynamics : application to hematopoiesis (DRACULA), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Modélisation mathématique, calcul scientifique (MMCS)

Subjects
Differential equation, General Mathematics, 010102 general mathematics, Mathematical analysis, General Engineering, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010103 numerical & computational mathematics, 0101 mathematics, 01 natural sciences, Laplace operator, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

International audience

Published
2020

28. On the Well-posedness of the Magnetic, Semi-relativistic Schrodinger-Poisson System

Author

Vitali Vougalter

Subjects
Physics, Control and Optimization, Computational Mechanics, Statistical and Nonlinear Physics, Kinetic energy, Strong solutions, symbols.namesake, symbols, Coulomb, Discrete Mathematics and Combinatorics, Uniqueness, Poisson system, Schrödinger's cat, Well posedness, Mathematical physics
Abstract

We prove global existence and uniqueness of strong solutions for the Schrodinger-Poisson system in the repulsive Coulomb case with relativistic, magnetic kinetic energy.

Published
2018

29. Existence in the Sense of Sequences of Stationary Solutions for Some Non-Fredholm Integro-Differential Equations

Author

Vitaly Volpert and Vitali Vougalter

Subjects
Statistics and Probability, Property (philosophy), Differential equation, Applied Mathematics, General Mathematics, 010102 general mathematics, Order (ring theory), Sense (electronics), Type (model theory), Differential operator, 01 natural sciences, 010101 applied mathematics, Convergence (routing), Bibliography, Applied mathematics, 0101 mathematics, Mathematics
Abstract

We establish the existence in the sense of sequences of stationary solutions for some reaction-diffusion type equations in appropriate H2 spaces. It is shown that, under reasonable technical conditions, the convergence in L1 of the integral kernels implies the existence and convergence in H2 of solutions. The nonlocal elliptic equations involve second order differential operators with and without the Fredholm property. Bibliography: 21 titles.

Published
2017

30. On the Existence of Stationary Solutions for Some Systems of Non-Fredholm Integro-Differential Equations with Superdiffusion

Author

Vitali Vougalter and Vitaly Volpert

Subjects
Control and Optimization, Property (philosophy), Differential equation, Mathematical analysis, Computational Mechanics, Discrete Mathematics and Combinatorics, Statistical and Nonlinear Physics, Fixed point, Vector-valued function, Mathematics
Abstract

We establish the existence of stationary solutions for certain systems of reaction-diffusion equations with superdiffusion. The corresponding elliptic problem involves the operators with or without Fredholm property. The fixed point technique in appropriate H2 spaces of vector functions is employed.

Published
2017

31. On the Well-posedness of the Magnetic Schrödinger-Poisson System in ℝ3

Author

Vitali Vougalter, Jean-Marie Barbaroux, CPT - E8 Dynamique quantique et analyse spectrale, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [University of Toronto], and University of Toronto

Subjects
density matrices, magnetic fields, 01 natural sciences, Schrödinger-Poisson system, Strong solutions, symbols.namesake, functional spaces, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Coulomb, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Mathematics, Applied Mathematics, 010102 general mathematics, global existence and uniqueness, Magnetic field, Classical mechanics, Modeling and Simulation, symbols, 010307 mathematical physics, Poisson system, Schrödinger's cat, Well posedness, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

International audience; We prove global existence and uniqueness of strong solutions for the Schrödinger-Poisson system in the repulsive Coulomb case in R 3 in the presence of a smooth magnetic field.

Published
2017

33. Existence of waves for a reaction–diffusion–dispersion system

Author

Moitri Sen, Vitaly Volpert, Vitali Vougalter, National Institute of Technology [Patna] (NIT), Multi-scale modelling of cell dynamics : application to hematopoiesis (DRACULA), Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan (ICJ), Department of Mathematics [University of Toronto], University of Toronto, Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Bistability, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Zero (complex analysis), Topological degree theory, System of linear equations, 01 natural sciences, 010101 applied mathematics, Nonlinear system, Reaction–diffusion system, A priori and a posteriori, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Dispersion (water waves), Analysis, Mathematics
Abstract

International audience; Existence of travelling waves is studied for a bistable reaction–diffusion system of equations with linear integral terms (dispersion) and with some conditions on the nonlinearity. The proof is based on the Leray–Schauder method using the topological degree theory for Fredholm and proper operators with the zero index and a priori estimates of solutions in properly chosen weighted spaces.

Published
2019

34. On the Solvability in the Sense of Sequences for Some Non-Fredholm Operators Related to the Anomalous Diffusion

Author

Vitali Vougalter and Vitaly Volpert

Subjects
Sobolev space, Physics, Pure mathematics, symbols.namesake, Anomalous diffusion, Convergence (routing), symbols, Order (ring theory), Scattering theory, Type (model theory), Differential operator, Schrödinger's cat
Abstract

We study solvability of some linear nonhomogeneous elliptic problems and prove that under reasonable technical conditions the convergence in \(L^{2}({\mathbb R}^{d})\) of their right sides implies the existence and the convergence in \(H^{2s}({\mathbb R}^{d})\) of the solutions. The equations involve the second order non-Fredholm differential operators raised to certain fractional powers s and we use the methods of spectral and scattering theory for Schrodinger type operators developed in our preceding work (Volpert and Vougalter, Electron J Differ Equ 160:16 pp, 2013).

Published
2019

35. Solvability of Some Integro-Differential Equations with Anomalous Diffusion in Two Dimensions

Author

Vitaly Volpert, Vitali Vougalter, Department of Mathematics [University of Toronto], University of Toronto, Multi-scale modelling of cell dynamics : application to hematopoiesis (DRACULA), Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan (ICJ), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Anomalous diffusion, Differential equation, Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, Fixed point, 01 natural sciences, Fractional power, Elliptic operator, 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 010306 general physics, Laplace operator, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We study the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in two dimensions. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.

Published
2018

39. On the Existence in the Sense of Sequences of Stationary Solutions for Some Systems of Non-Fredholm Integro-differential Equations

Author

Vitali Vougalter, Vitaly Volpert, Department of Mathematics [University of Toronto], University of Toronto, Multi-scale modelling of cell dynamics : application to hematopoiesis (DRACULA), Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan (ICJ), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
non-Fredholm operators, Pure mathematics, Property (philosophy), Differential equation, General Mathematics, 010102 general mathematics, Sense (electronics), Type (model theory), Differential operator, 01 natural sciences, 010101 applied mathematics, Solvability conditions, stationary solution, Convergence (routing), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, systems of integro-differential equations, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We prove the existence in the sense of sequences of stationary solutions for some systems of reaction–diffusion type equations in the appropriate $$H^{2}$$ spaces. It is established that, under reasonable technical conditions, the convergence in $$L^{1}$$ of the integral kernels yields the existence and the convergence in $$H^{2}$$ of the solutions. The nonlocal elliptic problems contain the second-order differential operators with and without Fredholm property.

Published
2018

40. Existence of stationary solutions for some integro-differential equations with anomalous diffusion

Author

Vitaly Volpert and Vitali Vougalter

Subjects
Elliptic operator, Partial differential equation, Anomalous diffusion, Differential equation, Applied Mathematics, Mathematical analysis, p-Laplacian, Laplace operator, Analysis, Sobolev spaces for planar domains, Mathematics, Sobolev inequality
Abstract

The article deals with the existence of solutions of an integro-differential equation arising in population dynamics in the case of anomalous diffusion involving the negative Laplace operator raised to a certain fractional power. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for non-Fredholm elliptic operators in unbounded domains along with the Sobolev inequality for a fractional Laplacian are being used.

Published
2015

41. Existence of stationary solutions for some nonlocal reaction-diffusion equations

Author

Vitali Vougalter, Vitaly Volpert, Department of Mathematics [University of Toronto], University of Toronto, Multi-scale modelling of cell dynamics : application to hematopoiesis (DRACULA), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), and Institut Camille Jordan (ICJ)

Subjects
education.field_of_study, Property (philosophy), Applied Mathematics, 010102 general mathematics, Mathematical analysis, Population, MathematicsofComputing_NUMERICALANALYSIS, Fredholm integral equation, Fixed point, 01 natural sciences, Fredholm theory, 010101 applied mathematics, Sobolev space, symbols.namesake, Elliptic operator, Reaction–diffusion system, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, education, Analysis, Mathematics
Abstract

International audience; The paper is devoted to the existence of solutions of a nonlocal reaction-diffusion equation arising in population dynamics. The proof is based on a fixed point technique. We use solvability conditions for elliptic operators in unbounded domains which do not satisfy the Fredholm property. CONTENTS

Published
2015

42. On the existence of stationary solutions for some nonlinear heat equations

Author

Vitali Vougalter, Vitaly Volpert, Multi-scale modelling of cell dynamics : application to hematopoiesis (DRACULA), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [University of Toronto], University of Toronto, Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), and Institut Camille Jordan (ICJ)

Subjects
General Mathematics, 010102 general mathematics, Mathematical analysis, Fixed point, Differential operator, 01 natural sciences, Fredholm theory, Nonlinear system, symbols.namesake, 0103 physical sciences, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Order (group theory), Heat equation, 0101 mathematics, 010306 general physics, Mathematics
Abstract

International audience; We establish the existence in H 3 (R 5) of stationary solutions of certain nonlinear heat equations using the Fixed Point Technique. The equation for the perturbed solution involves the second order differential operator without Fredholm property.

Published
2015

43. Solvability of Some Integro-Differential Equations with Anomalous Diffusion

Author

Vitali Vougalter and Vitaly Volpert

Subjects
Property (philosophy), Differential equation, Anomalous diffusion, 010102 general mathematics, Mathematical analysis, Fixed point, 01 natural sciences, Fractional power, Elliptic operator, 0103 physical sciences, 0101 mathematics, 010306 general physics, Laplace operator, Mathematics
Abstract

The paper deals with the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the Laplace operator in a fractional power. The proof of existence of solutions relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used.

Published
2017

44. Doubly nonlocal reaction-diffusion equations and the emergence of species

Author

Vitaly Volpert, Malay Banerjee, Vitali Vougalter, Department of Mathematics and Statistics [Kanpur], Indian Institute of Technology Kanpur (IIT Kanpur), Department of Mathematics [University of Toronto], University of Toronto, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL)

Subjects
Work (thermodynamics), Traveling waves, Population, Space (mathematics), Reaction–diffusion equation, 01 natural sciences, 010305 fluids & plasmas, 0103 physical sciences, Genetic algorithm, Reaction–diffusion system, Stationary pulses, Traveling wave, Quantitative Biology::Populations and Evolution, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, education, Mathematics, education.field_of_study, Nonlocal reproduction, Applied Mathematics, Mathematical analysis, 010101 applied mathematics, Nonlinear system, Homogeneous, Modeling and Simulation, Emergence of species
Abstract

International audience; The paper is devoted to a reaction-diffusion equation with doubly nonlocal nonlinearity arising in various applications in population dynamics. One of the integral terms corresponds to the nonlocal consumption of resources while another one describes reproduction with different phenotypes. Linear stability analysis of the homogeneous in space stationary solution is carried out. Existence of travelling waves is proved in the case of narrow kernels of the integrals. Periodic travelling waves are observed in numerical simulations. Existence of stationary solutions in the form of pulses is shown, and transition from periodic waves to pulses is studied. In the applications to the speciation theory, the results of this work signify that new species can emerge only if they do not have common offsprings. Thus, it is shown how Darwin's definition of species as groups of morphologically similar individuals is related to Mayr's definition as groups of individuals that can breed only among themselves.

Published
2017

45. Existence of stationary solutions for some systems of integro-differential equations with superdiffusion

Author

Vitaly Volpert, Vitali Vougalter, Department of Mathematics [University of Toronto], University of Toronto, Multi-scale modelling of cell dynamics : application to hematopoiesis (DRACULA), Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan (ICJ), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Integro-differential equations, Property (philosophy), Anomalous diffusion, General Mathematics, Population, Fixed point, 01 natural sciences, Fredholm theory, 010305 fluids & plasmas, symbols.namesake, 35J05, 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 47F05, education, Mathematics, education.field_of_study, 010102 general mathematics, Sobolev space, Elliptic operator, 35P30, Sobolev spaces, symbols, non Fredholm operators
Abstract

International audience; In this article, we establish the existence of solutions of a system of integro-differential equations arising in population dynamics in the case of anomalous diffusion. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without the Fredholm property in unbounded domains are used.

Published
2017

46. On the Solvability of Some Systems of Integro-Differential Equations with Anomalous Diffusion

Author

Vitali Vougalter and Vitaly Volpert

Subjects
Anomalous diffusion, Differential equation, 010102 general mathematics, Mathematical analysis, Fixed point, 01 natural sciences, Fractional power, Sobolev space, Elliptic operator, 0103 physical sciences, 0101 mathematics, 010306 general physics, Laplace operator, Mathematics
Abstract

The article deals with the existence of solutions of a system of integro-differential equations in the case of anomalous diffusion with the Laplacian in a fractional power. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.

Published
2017

47. Solvability in the sense of sequences for some non Fredholm operators related to the superdiffusion

Author

Vitaly Volpert, Vitali Vougalter, Department of Mathematics [University of Toronto], University of Toronto, Multi-scale modelling of cell dynamics : application to hematopoiesis (DRACULA), Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de génétique et de physiologie moléculaire et cellulaire (CGPhiMC), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discrete mathematics, Pure mathematics, Partial differential equation, Functional analysis, Applied Mathematics, 010102 general mathematics, Order (ring theory), Operator theory, Type (model theory), Differential operator, 01 natural sciences, 010101 applied mathematics, Sobolev space, AMS Subject Classification: 35J10, 35P10, 47F05, Sobolev spaces, solvability conditions, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Scattering theory, non Fredholm operators, 0101 mathematics, Analysis, Mathematics
Abstract

We study solvability of some linear nonhomogeneous elliptic equations and show that under reasonable technical conditions the convergence in \(L^{2}({{\mathbb {R}}}^{d})\) of their right sides yields the existence and the convergence in \(H^{1}({{\mathbb {R}}}^{d})\) of the solutions. The problems involve the square roots of the second order non Fredholm differential operators and we use the methods of spectral and scattering theory for Schrodinger type operators similarly to our preceding work (Volpert and Vougalter in Electron J Differ Equ 160:16, 2013).

Published
2017

48. Existence and nonlinear stability of stationary states for the magnetic Schrödinger-Poisson system

Author

Vitali Vougalter, Jean-Marie Barbaroux, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - E8 Dynamique quantique et analyse spectrale, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [University of Toronto], and University of Toronto

Subjects
Statistics and Probability, Physics, Applied Mathematics, General Mathematics, Nonlinear stability, 010102 general mathematics, Plasma, 01 natural sciences, Magnetic field, symbols.namesake, Classical mechanics, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Coulomb, symbols, 010307 mathematical physics, 0101 mathematics, Poisson system, [MATH]Mathematics [math], Schrödinger's cat, Stationary state
Abstract

International audience; The article is devoted to the studies of the stationary states of the magnetic Schrödinger-Poisson system in the repulsive (plasma physics) Coulomb case. Particularly, we prove the existence and the nonlinear stability of a wide class of stationary states by virtue of the energy-Casimir method. We generalize the global well-posedness result for the Schrödinger-Poisson system obtained by Brezzi and Markowich to the case when a magnetic field is turned on.

Published
2016

50. Existence and Nonlinear Stability of Stationary States for the Semi-Relativistic Schrödinger-Poisson System

Author

Vitali Vougalter, Thomas Chen, and Walid K. Abou Salem

Subjects
Nuclear and High Energy Physics, Class (set theory), Nonlinear stability, 010102 general mathematics, Mathematical analysis, Mathematics::Analysis of PDEs, Statistical and Nonlinear Physics, Plasma, Mathematics::Spectral Theory, 01 natural sciences, symbols.namesake, 82D10, 82C10, 0103 physical sciences, symbols, Coulomb, 010307 mathematical physics, 0101 mathematics, Poisson system, Mathematical Physics, Schrödinger's cat, Stationary state, Mathematics, Mathematical physics
Abstract

We study the stationary states of the semi-relativistic Schr\"odinger-Poisson system in the repulsive (plasma physics) Coulomb case. In particular, we establish the existence and the nonlinear stability of a wide class of stationary states by means of the energy-Casimir method. We generalize the global well-posedness result of our previous work for the semi-relativistic Schr\"odinger-Poisson system to spaces with higher regularity., Comment: Latex, 25 pages

Published
2013

Catalog

Books, media, physical & digital resources
See catalog results