Abstract:
Many questions related to well-posedness/ill-posedness in critical spaces for hydrodynamic equations have been open for many years. Some of them have only recently been settled. In this article we give a new approach to studying norm inflation (in some critical spaces) for a wide class of equations arising in hydrodynamics. As an application, we prove strong ill-posedness of the $d$-dimensional Euler equations in the class $C^1\cap L^2.$ We give two proofs of this result in sections 8 and 9.

Abstract:
A general exact theory of autoresonance (self-sustained resonance) in both dissipative and Hamiltonian nonautonomous systems is presented. The equations that together govern the autoresonance solutions and excitations are derived with the aid of a variational principle concerning the power functional. The theory is applied to Duffing oscillators to obtain exact analytical expressions for autoresonance excitations and solutions which explain all the phenomenological and approximate results arising from the previous approach to autoresonance phenomena.

Abstract:
We give an exhaustive characterization of singular weak solutions for ordinary differential equations of the form $\ddot{u}\,u + \frac{1}{2}\dot{u}^2 + F'(u) =0$, where $F$ is an analytic function. Our motivation stems from the fact that in the context of hydrodynamics several prominent equations are reducible to an equation of this form upon passing to a moving frame. We construct peaked and cusped waves, fronts with finite-time decay and compact solitary waves. We prove that one cannot obtain peaked and compactly supported traveling waves for the same equation. In particular, a peaked traveling wave cannot have compact support and vice versa. To exemplify the approach we apply our results to the Camassa-Holm equation and the equation for surface waves of moderate amplitude, and show how the different types of singular solutions can be obtained varying the energy level of the corresponding planar Hamiltonian systems.

Abstract:
In this paper, we prove the partial linearization for $n$-dimensional nonautonomous differential equations. The gap conditions are formulated in terms of the dichotomy spectral intervals.

Abstract:
By means of a direct and constructive method based on the theory of semiglobal $C^2$ solution, the local exact boundary observability is shown for nonautonomous 1-D quasilinear wave equations. The essential difference between nonautonomous wave equations and autonomous ones is also revealed.

Abstract:
As is well-known that the general radiation hydrodynamics models include two mainly coupled parts: one is macroscopic fluid part, which is governed by the compressible Navier-Stokes-Fourier equations, another is radiation field part, which is described by the transport equation of photons. Under the two physical approximations: "gray" approximation and P1 approximation, one can derive the so-called Navier-Stokes-Fourier-P1 approximation radiation hydrodynamics model from the general one. In this paper we study the non-relativistic limit problem for the Navier-Stokes-Fourier-P1 approximation model due to the fact that the speed of light is much larger than the speed of the macroscopic fluid. Our results give a rigorous derivation of the widely used macroscopic model in radiation hydrodynamics.

Abstract:
The upper semicontinuity and continuity properties of pullback attractors for nonautonomous differential equations are investigated when the driving system of the generated skew-product flow is digitized.

Abstract:
Nonlinear nonautonomous differential systems with delaying argument are considered. Explicit conditions for absolute stability are derived. The proposed approach is based on the generalization of the freezing method for ordinary differential equations.

Abstract:
The problem of integrable discretization of Liouville type hyperbolic PDE is studied. The method of discretization preserving characteristic integrals is adopted to nonautonomous case. An intriguing relation between Darboux integrable differential-difference equations and the Guldberg-Vessiot-Lie problem of describing all ODE, possessing fundamental solution systems is observed.