Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics

Zvyagin, V. G.; Vorotnikov, Dmitry

doi:10.1515/9783110208283

Cited by 81 publications

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“…Such triples (X, Y, X * ) are called Hilbert triples (sometimes also referred to as Gelfand or Lions triples), see, e.g., [18,20] for more details.…”

Section: Disgression: On Solvability Of Some Nonlinear Cauchy Problemmentioning

confidence: 99%

On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing I: Analysis

Prasath¹,

Vorotnikov²

2018

Nonlinear Analysis

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Variable exponent spaces have found interesting applications in real world problems. Recently, there have been considerable interest in utilizing variational and evolution problems based on variable exponents for imaging applications. The main classes of partial differential equations (PDEs) related to the variable exponents involve the p(·)-Laplacian. In imaging applications, the variable exponent can approach the critical value 1, and this poses unique challenges in proving existence of solutions, which have not been mastered earlier.In this work, we develop some additional functional framework to study the time-dependent parabolic flows with critical variable exponents. Specifically, we consider bounded vectorial partial variation (BVPV) space and its variable exponent counterpart. We prove the existence of weak solutions of critical vectorial p(t, x)-Laplacian flow in our variable exponent space. For non-timedependent variable exponent based critical vectorial p(x)-Laplacian flow we obtain a semigroup solution. The results are new even in the scalar case. This is a theory-oriented draft, and the full paper will provide detailed experimental results on color image restoration using various example for the variable exponents and compare them traditional PDE based image processing procedures. Our results will indicate the applicability of variable exponent Laplacian flows in image processing in general and image restoration in particular.

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“…Such triples (X, Y, X * ) are called Hilbert triples (sometimes also referred to as Gelfand or Lions triples), see, e.g., [18,20] for more details.…”

Section: Disgression: On Solvability Of Some Nonlinear Cauchy Problemmentioning

confidence: 99%

On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing I: Analysis

Prasath¹,

Vorotnikov²

2018

Nonlinear Analysis

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“…Let us now recall some definitions and results from [47,52]. Let E ⊂ E 0 be Banach spaces with E reflexive.…”

Section: Vol 3 (2008)mentioning

confidence: 99%

“…Theorem 6.2 (see [52]). Assume that there exists a uniform trajectory semiattractor P for equation (6.4).…”

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confidence: 99%

Approximating-topological methods in some problems of hydrodynamics

Zvyagin

Vorotnikov

2008

J. fixed point theory appl.

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We describe a new approach to the study of initial-boundary value problems for evolutionary equations of hydrodynamics, which is based on approximation of the problems and subsequent application of topological degree theory for investigation of weak solvability for approximating problems. Use of this method turned out to be especially effective in problems of nonNewtonian hydrodynamics. We demonstrate its application to the study of weak solvability and attractors of the initial-boundary value problem for the Oldroyd model with regularized objective Jaumann derivative. (2000). Primary 35Q35; Secondary 47H11. Mathematics Subject Classification

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“…Разрешимость в слабом смысле начально-краевой задачи (0.1)-(0.5) уста-новлена в работах [4], [5] и подробно описана в [6]. Следующим по важности обычно является описание поведения решения задач при больших значениях времени.…”

Section: Introductionunclassified

“…свойства используемых операторов и оценка решений В статье используется аппроксимационно-топологический метод исследова-ния уравнений, предложенный в [14] и развитый в [6]. Исходя из этого метода введем аппроксимационные уравнения, используемые в задаче (0.1)-(0.5).…”

Section: Introductionunclassified

Аттракторы Слабых Решений Регуляризованной Системы Уравнений Движения Жидких Сред С Памятью

Звягин¹,

Zvyagin²,

Kondrat'ev³

2012

Матем. сб.

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Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics

Cited by 81 publications

References 0 publications

On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing I: Analysis

On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing I: Analysis

Approximating-topological methods in some problems of hydrodynamics

Аттракторы Слабых Решений Регуляризованной Системы Уравнений Движения Жидких Сред С Памятью

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