Mathematical Problems from Combustion Theory

Bebernes, J.; Eberly, David H.

doi:10.1007/978-1-4612-4546-9

Cited by 492 publications

(455 citation statements)

References 13 publications

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“…This equation is a simple model for combustion [5], and it is well known (see e.g. [27]) that if u 0 is sufficiently large then the solution u will become unbounded in finite time.…”

Section: Selecting the Monitor Function For The Dynamic Meshmentioning

confidence: 99%

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Ceniceros¹,

Hou²

2001

Journal of Computational Physics

147

156

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We develop an efficient dynamically adaptive mesh generator for time-dependent problems in two or more dimensions. The mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic PDEs for the mesh map. When coupled to a physical problem, the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior. The overall mesh strategy is simple to implement, avoids interpolation, and can be easily incorporated into a broad range of applications. The efficacy of the mesh is first demonstrated by two examples of blowing-up solutions to the 2-D semilinear heat equation. These examples show that the mesh can follow with high adaptivity a finite-time singularity process. The focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2-D Boussinesq fluid, a challenging problem. The moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically. The numerical results show the fast collapse to small scales and an exponential vorticity growth.

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“…This equation is a simple model for combustion [5], and it is well known (see e.g. [27]) that if u 0 is sufficiently large then the solution u will become unbounded in finite time.…”

Section: Selecting the Monitor Function For The Dynamic Meshmentioning

confidence: 99%

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Ceniceros¹,

Hou²

2001

Journal of Computational Physics

147

156

View full text Add to dashboard Cite

show abstract

“…More recent reviews include (Levine, 1990) and (Deng and Levine, 2000). Exponentially growing functions, which also lead to blow up, have been considered (Gelfand, 1963;Bebernes and Eberly, 1989) because their importance in some combustion problems. The analysis leading to blow up results usually makes use of differential inequalities and the maximum principle.…”

Section: Blow Up Phenomena In Nonlinear Parabolic Equationsmentioning

confidence: 99%

Volterra boundary control laws for a class of nonlinear parabolic partial differential equations

Vázquez¹,

Krstić

2004

IFAC Proceedings Volumes

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The efforts on boundary control of general classes of nonlinear parabolic PDEs with nonlinearities of superlinear growth have so far only resulted in counterexamples-results that show that finite time blow up occurs for large initial conditions even for simple cases like quadratic nonlinearities, with or without control. In this paper we present results identifying a class of systems that is stabilizable. Our approach is a direct infinite dimensional extension of the feedback linearization/backstepping approach and employs Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. While the full detail of our general approach is left for a future publication without a page limit, in this paper we give an example with explicit solutions for the plant/controller pair, including an explicit construction of the inverse of the feedback linearizing Volterra transformation. This, in turn, allows us to explicitly construct the exponentially decaying closed loop solutions. We include also a numerical illustration, showing blow up in open loop, and stabilization for large initial conditions in closed loop.

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“…If the magnitude of the solution is diverging, |u| → ∞, the behavior is generally called finite-time blow-up. Blow-up has been studied extensively in reaction-diffusion equations in combustion theory [6] and mathematical biology [66]. Problems having the time-derivative of the solution diverging, |u t | → ∞, while the solution remains bounded are said to exhibit quenching [54].…”

Section: Introductionmentioning

confidence: 99%

“…In the pinch-off of a thin liquid filament, as in a lava lamp or the break-up of a fluid jet, the profile of the pinching neck assumes a universal shape as detachment is approached [18,32]. Other examples include rupture of a solid filament due to surface diffusion [11], rupture of thin films [79,69,70,75,76], and the ignition of a combustible substance [6].…”

Section: Introductionmentioning

confidence: 99%

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Stability and dynamics of self-similarity in evolution equations

Bernoff

Witelski

2009

J Eng Math

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Abstract. We discuss a methodology for studying the linear stability of self-similar solutions. We will illustrate these fundamental ideas on three prototype problems: a simple ODE with finite-time blowup, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, we will discuss the use of dimensional analysis to derive scaling invariant similarity variables, and the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions we will demonstrate that the symmetries give rise to positive eigenvalues associated with the symmetries, and show how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions.

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Mathematical Problems from Combustion Theory

Cited by 492 publications

References 13 publications

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Volterra boundary control laws for a class of nonlinear parabolic partial differential equations

Stability and dynamics of self-similarity in evolution equations

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