Blow up and regularity for fractal Burgers equation
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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.Abstract:
The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove 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. We also prove the existence of solutions with very rough initial data u0 ∈ Lp, 1 < p < ∞. Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation.read more
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References
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TL;DR: In this article, an introduction to vortex dynamics for incompressible fluid flows is given, along with vortex sheets, weak solutions and approximate-solution sequences for the Euler equation.
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Navier-Stokes equations
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TL;DR: Navier-Stokes Equations as mentioned in this paper provide a compact and self-contained course on these classical, nonlinear, partial differential equations, which are used to describe and analyze fluid dynamics and the flow of gases.
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TL;DR: In this paper, the formation of strong and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied through the symbiotic interaction of mathematical theory and numerical experiments.
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A Maximum Principle Applied to Quasi-Geostrophic Equations
Antonio Córdoba,Diego Córdoba +1 more
TL;DR: In this paper, the authors study the initial value problem for dissipative 2D quasi-geostrophic equations proving local existence, global results for small initial data in the super-critical case, decay of Lp-norms and asymptotic behavior of viscosity solution in the critical case.