The inflow problem of full compressible NavierStokes equations is considered on the half-line (0,+\infty). First, we give the existence (or nonexistence) of the boundary layer solution to the inflow problem when the right end state (0,+\infty) belongs to the subsonic, transonic, and supersonic regions, respectively. Then the asymptotic stability of not only the single contact wave but also the superposition of the subsonic boundary layer solution, the contact wave, and the rarefaction wave to the inflow problem are investigated under some smallness conditions. Note that the amplitude of the rarefaction wave is not necessarily small. The proofs are given by the elementary energy method.

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We establish local higher integrability estimates for upper gradients of vector-valued parabolic quasi-minimizers in metric measure spaces, satisfying a doubling property and supporting a weak Poincaré inequality.

In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $\bu = \n^\perp \Psi$, $\Psi(r,\th) = r^{\l} \psi(\th)$, for $\l >0$, we show that only trivial solutions exist in the range $0<\l<1/2$, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for $\l>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2\l}) \cap \N$. The case $\l = 2/3$ is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity $1/3$.

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