On global solutions and blow-up for Kuramoto–Sivashinsky-type models, and well-posed Burnett equations

Galaktionov, Victor A.; Mitidieri, Enzo; Pohožaev, Stanislav I.

doi:10.1016/j.na.2008.12.020

Cited by 30 publications

(26 citation statements)

References 65 publications

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“…(1 + ρ (2) ), becomes more negative with increasing ρ (2) , which is known to give rise to singularities developing in a finite time (e.g., [25,26]). …”

Section: A the Analysismentioning

confidence: 99%

Semiphenomenological model for gas-liquid phase transitions

Benilov

2016

Phys. Rev. E

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We examine a rarefied gas with inter-molecular attraction. It is argued that the attraction force amplifies random density fluctuations by pulling molecules from lower-density regions into high-density regions and thus may give rise to an instability. To describe this effect, we use a kinetic equation where the attraction force is taken into account in a way similar to how electromagnetic forces in plasma are treated in the Vlasov model. It is demonstrated that the instability occurs when the temperature T is lower than a certain threshold value T s depending on the gas density. It is further shown that, even if T is only marginally lower than T s , the instability generates clusters with density much higher than that of the gas. These results suggest that the instability should be interpreted as a gas-liquid phase transition, with T s being the temperature of saturated vapor and the high-density clusters representing liquid droplets.

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“…(1 + ρ (2) ), becomes more negative with increasing ρ (2) , which is known to give rise to singularities developing in a finite time (e.g., [25,26]). …”

Section: A the Analysismentioning

confidence: 99%

Semiphenomenological model for gas-liquid phase transitions

Benilov

2016

Phys. Rev. E

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“…This crucial question remains open for three-dimensional inviscid flows governed by the Euler equations, as well as for three-dimensional viscous flows of Newtonian fluids. The general topic of blow-ups in nonlinear systems is, in fact, an area of active research [1]. We argue below that the physical origin of the difficulty in reaching an answer involves the interplay between the nonlinear and the scale-invariant aspects of hydrodynamics.…”

Section: T M Viswanathan and G M Viswanathanmentioning

confidence: 98%

Hydrodynamics at the smallest scales: a solvability criterion for Navier–Stokes equations in high dimensions

Viswanathan

2011

Phil. Trans. R. Soc. A.

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Strong global solvability is difficult to prove for high-dimensional hydrodynamic systems because of the complex interplay between nonlinearity and scale invariance. We define the Ladyzhenskaya-Lions exponent a L (n) = (2 + n)/4 for Navier-Stokes equations with dissipation −(−D) a in R n , for all n ≥ 2. We review the proof of strong global solvability when a ≥ a L (n), given smooth initial data. If the corresponding Euler equations for n > 2 were to allow uncontrolled growth of the enstrophy (1/2) Vu 2 L 2 , then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for a < a L (n). The energy is critical under scale transformations only for a = a L (n).

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“…Global unsolvability. To obtain sufficient conditions of the global unsolvability, we use the nonlinear capacity method (method of the test functions), suggested by Lions , Baras, Pierre , Pokhozhaev, Mitidiery, and Galaktionov ). This method is well studied and can be used both for classical and weak solutions.…”

Section: Analytical Approachmentioning

confidence: 99%

Blow‐up phenomena in the model of a space charge stratification in semiconductors: analytical and numerical analysis

Корпусов

Lukyanenko

Panin

et al. 2016

Math Methods in App Sciences

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The initial‐boundary value problems for a Sobolev equation with exponential nonlinearities, classical, and nonclassical boundary conditions are considered. For this model, which describes processes in crystalline semiconductors, the blow‐up phenomena are studied. The sufficient blow‐up conditions and the blow‐up time are analyzed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow‐up more accurately. The model derivation and some questions of local solvability and uniqueness are also discussed. Copyright © 2016 John Wiley & Sons, Ltd.

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On global solutions and blow-up for Kuramoto–Sivashinsky-type models, and well-posed Burnett equations

Cited by 30 publications

References 65 publications

Semiphenomenological model for gas-liquid phase transitions

Semiphenomenological model for gas-liquid phase transitions

Hydrodynamics at the smallest scales: a solvability criterion for Navier–Stokes equations in high dimensions

Blow‐up phenomena in the model of a space charge stratification in semiconductors: analytical and numerical analysis

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