Occurrence and non-appearance of shocks in fractal burgers equations
TLDR
In this paper, the fractal Burgers equation is considered and it is shown that, if the power of the Laplacian involved is lower than 1/2, then the equation does not regularize the initial condition.Abstract:
We consider the fractal Burgers equation (that is to say the Burgers equation to which is added a fractional power of the Laplacian) and we prove that, if the power of the Laplacian involved is lower than 1/2, then the equation does not regularize the initial condition: on the contrary to what happens if the power of the Laplacian is greater than 1/2, discontinuities in the initial data can persist in the solution and shocks can develop even for smooth initial data. We also prove that the creation of shocks can occur only for sufficiently "large" initial conditions, by giving a result which states that, for smooth "small" initial data, the solution remains at least Lipschitz continuous.read more
Citations
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Journal ArticleDOI
Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian
TL;DR: In this article, the existence of positive solutions for the nonlinear Schrodinger equation with the fractional Laplacian was studied and the regularity, decay and symmetry properties of these solutions were analyzed.
Journal ArticleDOI
Calcul des Probabilités. By Paul Lévy. Pp. 350. Fr. 40. 1925. (Gauthier-Villars.)
Journal ArticleDOI
Blow up and regularity for fractal Burgers equation
TL;DR: In this paper, the authors studied the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation, and proved the existence of the finite time blow up for the power of Laplacian α < 1/2, and global existence as well as analyticity of solution for α ≥ 1 2.
Journal ArticleDOI
Finite time blowup for an averaged three-dimensional Navier-Stokes equation
TL;DR: In this article, a modified version of the Navier-stokes global regularity problem has been studied in three dimensions, where the average involves rotations and Fourier multipliers of order zero.
Posted Content
Discontinuous Galerkin method for fractional convection-diffusion equations
Qinwu Xu,Jan S. Hesthaven +1 more
TL;DR: In this paper, a discontinuous Galerkin method for convection-subdiffusion equations with a fractional operator of order α (1 < α < 2) defined through the fractional Laplacian is proposed.
References
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An Introduction to Probability Theory and Its Applications
David A. Freedman,William Feller +1 more
An Introduction To Probability Theory And Its Applications
TL;DR: A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.
Journal ArticleDOI
Calcul des Probabilités. By Paul Lévy. Pp. 350. Fr. 40. 1925. (Gauthier-Villars.)
Book
Global classical solutions for quasilinear hyperbolic systems
TL;DR: Cauchy Problem for Single First Order Equations Cauchy problem for Reducible Quasilinear Hyperbolic Systems as mentioned in this paper for general QH systems with Dissipation Mixed Initial-Boundary Value Problem with Boundary Dissipation for QuasILBolic Systems Typical Boundary value Problem and Typical Free Boundary Problem for RedUCible QuASILINear HSBs Generalized Riemann Problem for the System of One-Dimensional Isentropic Flow Typical free boundary problem for General QHSBs Bibliography Index.