In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data. In particular, the results hold true for the full compressible Euler equations and isentropic compressible Euler equations and the blow-up time can be computed in a more precise way. It is not required that the initial data has compact support or contains vacuum in any finite regions. Moreover, we will give a simplified and unified proof on the blow-up results to the classical solutions of the full compressible Navier-Stokes equations without heat conduction by Xin [41] and with heat conduction by Cho-Jin [5].

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This work concerns the local well-posedness to the Cauchy problem of a fully dispersive Boussinesq system which models fully dispersive water waves in two and three spatial dimensions. Our purpose is to understand the modified energy approach (Kalisch and Pilod in Proc Am Math Soc 147:2545-2559, 2019) in a different point view by utilizing the symmetrization of hyperbolic systems which produces an equivalent modified energy. Keywords Well-posedness • Dispersive Boussinesq system • Symmetrization Mathematics Subject Classification 76B15 • 76B03 • 35S30 in which t ∈ R, x = (x 1 , x 2) ∈ R 2 and the unknowns η ∈ R, v = (v 1 , v 2) ∈ R 2. The operator K is a Fourier multiplier with symbol m ∈ S a ∞ (R d), d = 1 or 2 for some a ∈ R\{0}, Y.W. acknowledges the support by Grants Nos. 231668 and 250070 from the Research Council of Norway.

We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.

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