Qualitative Theory of Parabolic Equations, Part 1

Zelenyak, T. I.; Vishnevskiî, M. P.; Lavrentiev, Mikhail

doi:10.1515/9783110935042

Cited by 18 publications

(12 citation statements)

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“…Finally, we note that one can also use the Lyapunov exponent associated with the parameter α to obtain additional information about the long time sensitivity of the solutions to Burgers' equation and other similar systems. Space does not allow for a full discussion of this topic, but the authors have been able to exploit the theory of Lyapunov exponents to obtain additional bounds on the sensitivities for coupled systems of parabolic problems of the type considered in [19]. …”

Section: Discussionmentioning

confidence: 99%

Control and sensitivity reduction for a viscous Burgers' equation

Allen

Burns

Gilliam

2012

2012 IEEE 51st IEEE Conference on Decision and Control (CDC)

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The problem of designing and numerically implementing a controller for a fluid flow system at the boundary of the flow domain is complicated by the facts that the basic model is truly nonlinear and the flow is usually highly sensitive to boundary conditions. High sensitivity to small changes in the boundary such as wall roughness or dynamic excitations can trigger transition to undesirable states. One approach to preventing or delaying a transition is to introduce a simple control loop along the boundary to reduce the sensitivity. In this work we demonstrate the aforementioned idea for a system governed by the one dimensional Burgers' equation. In particular, we focus on the initial boundary value problem for a viscous Burgers' equation which is known to be is extremely sensitive with respect to a small perturbation in a Neumann boundary condition. We use this model to illustrate how this sensitivity can generate erroneous numerical solutions and "transitions" to these states. In particular, for a fixed viscosity and certain initial data the numerical solution z(x, t) of Burgers' equation with a Neumann boundary condition zx(0, t) = 0 converges to a (large) solution that satisfies zx(0, t) = −α for a number α > 0 less than machine precision zero. Thus, the solution of the Burgers' equations for this problem exhibits an extreme sensitivity to the boundary perturbation α and this sensitivity can produce unexpected and undesired dynamic behavior. We use a continuous sensitivity equation method to compute these sensitivities and show that the sensitivity can be eliminated by introducing a very simple proportional error boundary feedback mechanism.

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Section: Discussionmentioning

confidence: 99%

Control and sensitivity reduction for a viscous Burgers' equation

Allen

Burns

Gilliam

2012

2012 IEEE 51st IEEE Conference on Decision and Control (CDC)

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“…Further extended gradient systems. As noted by Zelenyak [45], and extended by Matano, Fiedler, Poláčik, Rocha and others ( [14], [28] and references therein), there is a number of examples of nonlinearities g with a Lyapunov function on the bounded domain (with periodic or other boundary conditions). The discussion in Subsection 13.3 thus naturally leads to the following: Problem (4).…”

Section: 2mentioning

confidence: 99%

Ergodic attractors and almost-everywhere asymptotics of scalar semilinear parabolic differential equations

Slijepčević

2018

Journal of Differential Equations

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We consider dynamics of scalar semilinear parabolic equations on bounded intervals with periodic boundary conditions, and on the entire real line, with a general nonlinearity g(t, x, u, ux) either not depending on t, or periodic in t. While the topological and geometric structure of their attractors has been investigated in depth, we focus here on ergodic-theoretical properties. The main result is that the union of supports of all the invariant measures projects one-to-one to R 2 . We rely on a novel application of the zero-number techniques with respect to evolution of measures on the phase space, and on properties of the flux of zeroes, and the dissipation of zeroes. As an example of an application, we prove uniqueness of an invariant measure for a large family of considered equations which conserve a certain quantity ("mass"), thus generalizing the results by Sinai for the scalar viscous Burgers equation.

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“…The issues related to the statement, solvability, and regularity properties of the quasilinear parabolic problems have been reported by many authors, the well-known monographs and surveys [3,[9][10][11][12]19,20,24,27,28] among them. The results on existence and uniqueness of weak solutions of the initial-boundary value problems for quasilinear parabolic equations in the general form are given in the monograph [11].…”

Section: Solvability Of the Initial-boundary Value Problemmentioning

confidence: 99%

The problem of data assimilation for soil water movement

Dimet¹,

Shutyaev

Wang

et al. 2004

ESAIM: COCV

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Abstract.The soil water movement model governed by the initial-boundary value problem for a quasilinear 1-D parabolic equation with nonlinear coefficients is considered. The generalized statement of the problem is formulated. The solvability of the problem is proved in a certain class of functional spaces. The data assimilation problem for this model is analysed. The numerical results are presented.Mathematics Subject Classification. 65K10.

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Qualitative Theory of Parabolic Equations, Part 1

Cited by 18 publications

References 0 publications

Control and sensitivity reduction for a viscous Burgers' equation

Control and sensitivity reduction for a viscous Burgers' equation

Ergodic attractors and almost-everywhere asymptotics of scalar semilinear parabolic differential equations

The problem of data assimilation for soil water movement

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