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Building on the work of Crouseilles and Faou on the 2D case, we construct C ∞ quasi-periodic solutions to the incompressible Euler equations with periodic boundary conditions in dimension 3 and in any even dimension. These solutions are genuinely high-dimensional, which is particularly interesting because there are extremely few examples of high-dimensional initial data for which global solutions are known to exist. These quasi-periodic solutions can be engineered so that they are dense on tori of arbitrary dimension embedded in the space of solenoidal vector fields. Furthermore, in the two-dimensional case we show that quasi-periodic solutions are dense in the phase space of the Euler equations. More precisely, for any integer N ⩾ 1 we prove that any L q initial stream function can be approximated in L q (strongly when 1 ⩽ q < ∞ and weak-⁎ when q = ∞) by smooth initial data whose solutions are dense on N -dimensional tori. [ABSTRACT FROM AUTHOR]

We show that the log-Hölder coefficients of a solution to the transport equation is preserved in time. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Rayleigh-Taylor instability of incompressible Euler equations in a horizontal slab domain, which develops the results of Hwang and Guo (2003) in [11] by taking into account the boundary condition. If a steady density profile is non-monotonic, then the smooth steady state is nonlinearly unstable. Moreover, we also give a new proof for the local existence to inhomogeneous incompressible Euler equations in a smooth bounded domain Ω ⊂ R n , with initial data in H s (Ω) (s > n 2 + 1). [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the weak solutions of the three-dimensional incompressible Euler equations with helical symmetry in the whole space when the helical swirl vanishes. Specifically, we establish the global existence of weak solutions when the initial vorticity lies in L 1 ∩ L p with p > 1 . Our result extends the previous work [2] , where the initial vorticity is compactly supported and belongs to L p with p > 4 / 3 . The key ingredient in this paper involves the explicit analysis of Biot–Savart law with helical symmetry in domain R 2 × [ − π , π ] via the theories of singular integral operators and second order elliptic equations. [ABSTRACT FROM AUTHOR]

Abstract: The quasineutral limit of compressible Navier–Stokes–Poisson system with heat conductivity and general (ill-prepared) initial data is rigorously proved in this paper. It is proved that, as the Debye length tends to zero, the solution of the compressible Navier–Stokes–Poisson system converges strongly to the strong solution of the incompressible Navier–Stokes equations plus a term of fast singular oscillating gradient vector fields. Moreover, if the Debye length, the viscosity coefficients and the heat conductivity coefficient independently go to zero, we obtain the incompressible Euler equations. In both cases the convergence rates are obtained. [Copyright &y& Elsevier]