Relying on complex analytic methods we establish convexity and monotonicity properties of certain flow quantities associated with two-dimensional irrotational waves that propagate at the surface of water over a flat bed. Special attention is given to the case of solitary waves.
This paper investigates the three-dimensional stationary gas dynamic equations for an ideal adiabatic gas and employs symmetry-based methods to construct exact analytical solutions. In the Chaplygin gas case, the analysis produces a remarkably general family of solutions depending on three arbitrary functions, whereas for a general adiabatic index the equations admit several classes of explicit solutions parameterized by a finite set of constant parameters.
This paper concerns a cubic Camassa-Holm-type equation, a quasilinear partial differential equation with cubic nonlinearity derived as an asymptotic model for 2D full water wave dynamics. We focus on analyzing the wave-breaking phenomenon and establishing sufficient conditions on the initial data that guaranty finite-time blow-up of the solution.
We generalize our previous theorem for Friedmann-Lema & icirc;tre-Robertson-Walker (FLRW) spacetimes within the framework of generic metric gravity theories. In earlier work, we proved that, in the absence of additional matter fields, the field equations of metric gravity theories constructed from the curvature tensor and its covariant derivatives reduce in FLRW spacetime to an effective perfect-fluid form. In the present work, we extend this tensorial reduction to a class of local scalar-vector-tensor theories in which the gravitational action contains a homogeneous scalar field, a purely timelike isotropic vector field, and their covariant derivatives at arbitrary order. We prove that, under the FLRW-compatible assumptions phi = phi(t) and A mu = psi(t)u mu, the symmetric rank-two tensors entering the metric field equations necessarily reduce to the FLRW perfect-fluid tensorial sector. When an explicitly normalized Einstein-Hilbert term is included in the action, the metric field equations may therefore be written in Einstein form with an effective perfect-fluid source, supplemented by the corresponding scalar and vector field equations. This result should be understood as a background-level structural reduction, or as a perfect-fluid-type tensorial universality, rather than as a claim of strict metric universality. The effective density and pressure remain theory dependent, and therefore so does the resulting cosmological evolution. We illustrate the theorem using recently proposed Einstein-scalar and Einstein-Proca theories arising from regularized four-dimensional Einstein-Gauss-Bonnet gravity.
We investigate the large-time asymptotic behavior of solutions to the Cauchy problem for a one-dimensional nonlinear viscoelastic model with non-Newtonian type viscosity (μ1 + μ2|ux|p−1)ux (p > 1, which represents a dilatant fluid) and a non-strictly convex stress function. By adopting an energy method and introducing a cutoff function for domain decomposition, we prove the stability of the strong rarefaction wave under large initial perturbation, without requiring the smallness of wave strength, provided that μ2 is sufficiently small. In particular, we not only establish the Lk-stability (k∈2,+∞) of the strong rarefaction wave, but also show that the non-Newtonian viscous term yields an Lp+1 decay property for spatial derivative.
