  This is the first of a series of papers devoted to the initial value problem
for the Euler system of compressible fluids and augmented versions containing
higher-order terms. We encompass solutions that have finite total energy and
enjoy a certain symmetry (for instance, plane symmetry); these solutions may
have unbounded amplitude and contain cavitation regions in which the mass
density vanishes. In the present paper, we are interested in dispersive shock
waves and analyze the zero viscosity-capillarity limit associated with the
Navier-Stokes-Korteweg system. Specifically, we establish the existence of
finite energy solutions as well as their convergence toward entropy solutions
to the Euler system. We encompass a broad class of nonlinear
Navier-Stokes-Korteweg constitutive laws, which is determined by two main
conditions relating the viscosity and capillarity coefficients, that is, on one
hand the strong coercivity condition (as we call it) which provides a favorable
sign for the integrated dissipation associated with an effective energy, and on
the other hand the tame capillarity condition (as we call it), which restricts
pointwise the strength of the capillarity relatively to the viscosity. Rather
mild conditions on the growth of the constitutive functions are aso imposed,
which are required in order to define finite energy weak solutions to the
Navier-Stokes-Korteweg system, even in the presence of cavitation. Our method
of proof relies on fine algebraic properties of the Euler system and combines
together energy and effective energy estimates, dissipation and effective
dissipation estimates, a nonlinear Sobolev inequality, high-integrability
properties for the mass density and for the velocity, and compactness
properties based on entropies.
