Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global strong solution for the Korteweg system with quantum pressure in
  dimension $N\geq 2$   This work is devoted to prove the existence of global strong solution in
dimension $N\geq 2$ for a isothermal model of capillary fluids derived by J.E
Dunn and J.Serrin (1985) (see \cite{fDS}), which can be used as a phase
transition model. We will restrict us to the case of the so called compressible
Navier-Stokes system with quantum pressure which corresponds to consider the
capillary coefficient $\kappa(\rho)=\frac{\kappa_1}{\rho}$ with $\kappa_1>0$.
In a first part we prove the existence of strong solution in finite time for
large initial data with a precise bound by below on the life span $T^*$. This
one depends on the norm of the initial data $(\rho_0,v_0)$. The second part
consists in proving the existence of global strong solution with particular
choice on the capillary coefficient ( where $\kappa_1=\mu^2$) and on the
viscosity tensor which corresponds to the viscous shallow water case $-2\mu{\rm
div}(\rho Du)$. To do this we derivate different energy estimate on the density
and the effective velocity $v$ which ensures that the strong solution can be
extended beyond $T^*$. The main difficulty consists in controlling the vacuum
or in other words to estimate the $L^\infty$ norm of $\frac{1}{\rho}$. The
proof relies mostly on a method introduced by De Giorgi \cite{DG} (see also
Ladyzhenskaya et al in \cite{La} for the parabolic case) to obtain regularity
results for elliptic equations with discontinuous diffusion coefficients and a
suitable bootstrap argument.

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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an