Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Blowup Criterion for the Compressible Flows with Vacuum States   We prove that the maximum norm of the deformation tensor of velocity
gradients controls the possible breakdown of smooth(strong) solutions for the
3-dimensional compressible Navier-Stokes equations, which will happen, for
example, if the initial density is compactly supported \cite{X1}. More
precisely, if a solution of the compressible Navier-Stokes equations is
initially regular and loses its regularity at some later time, then the loss of
regularity implies the growth without bound of the deformation tensor as the
critical time approaches. Our result is the same as Ponce's criterion for
3-dimensional incompressible Euler equations (\cite{po}). Moreover, our method
can be generalized to the full Compressible Navier-Stokes system which improve
the previous results. In addition, initial vacuum states are allowed in our
cases.

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The categories of this paper are: cmp-lg
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The categories of this paper are: cs
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: econ
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The categories of this paper are: ao-sci
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin