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Well-posedness for the diffusive 3D Burgers equations with initial data in
In this note we discuss the diffusive, vector-valued Burgers equations in a
three-dimensional domain with periodic boundary conditions. We prove that given
initial data in these equations admit a unique global solution that
becomes classical immediately after the initial time. To prove local existence,
we follow as closely as possible an argument giving local existence for the
Navier--Stokes equations. The existence of global classical solutions is then a
consequence of the maximum principle for the Burgers equations due to Kiselev
and Ladyzhenskaya (1957).
In several places we encounter difficulties that are not present in the
corresponding analysis of the Navier--Stokes equations. These are essentially
due to the absence of any of the cancellations afforded by incompressibility,
and the lack of conservation of mass. Indeed, standard means of obtaining
estimates in fail and we are forced to start with more regular data.
Furthermore, we must control the total momentum and carefully check how it
impacts on various standard estimates.Comment: 15 pages, to appear in "Recent Progress in the Theory of the Euler
and Navier--Stokes Equations", eds. J.C. Robinson, J.L. Rodrigo, W. Sadowski
and A. Vidal-L\'opez, Cambridge University Press, 201
Effects of Various Combinations and Numbers of Lead: Iron Pellets Dosed in Wild-Type Captive Mallards
Final Report, Contract No. 14-16-0008-914INHS Technical Report prepared for unspecified recipien
The Navier-Stokes regularity problem
There is currently no proof guaranteeing that, given a smooth initial condition, the three-dimensional Navier–Stokes equations have a unique solution that exists for all positive times. This paper reviews the key rigorous results concerning the existence and uniqueness of solutions for this model. In particular, the link between the regularity of solutions and their uniqueness is highlighted
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