Related papers: Solvability of a class of evolution operators on c…

We consider the Cauchy problem for third-order evolution differential operators with variable coefficients, depending on time $t\in [0,T]$ and space $x\in\mathbb{R}$, where the leading coefficient $a_3(t)$ vanishes at $t = 0$ with finite…

In this work we begin a theoretical and numerical investigation on the spectra of evolution operators of neutral renewal equations, with the stability of equilibria and periodic orbits in mind. We start from the simplest form of linear…

In this paper, we establish two major classes of Liouville type results for the three-dimensional stationary tropical climate model. The first class is obtained under the assumptions imposed on $u,v,\theta$ whereas the second one relies on…

We consider the Lie algebra of derivations of a zero dimensional local complex algebra. We describe an inequality involving the embedding dimension, the order, and the first deviation that forces this Lie algebra to be solvable. Our result…

The question of complete integrability of evolution equations associated to $n\times n$ first order isospectral operators is investigated using the inverse scattering method. It is shown that for $n>2$, e.g. for the three-wave interaction,…

We generalize earlier results of Fokas and Liu and find all locally analytic (1+1)-dimensional evolution equations of order $n$ that admit an $N$-shock type solution with $N\leq n+1$. To this end we develop a refinement of the technique…

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs. In [arXiv:1303.3575], for any (1+1)-dimensional scalar…

Despite considerable developments in the literature of the past decades, a standing open problem in the analysis of continuum mechanics appears to consist of determining how far the prototypical model for small-strain thermoviscoelastic…

For a scalar evolution equation $u_t=K(t,x,u,u_x,\ldots, u_n), n\geq 2$ the cohomology spaces $H^{1,s}({\mathcal R}^\infty)$ vanishes for $s\geq 3$ while the space $H^{1,2}({\mathcal R}^\infty)$ is isomorphic to the space of variational…

Found are conditions on a scalar type spectral operator $A$ in a complex Banach space necessary and sufficient for all weak solutions of the evolution equation \begin{equation*} y'(t)=Ay(t),\ t\ge 0, \end{equation*} to be strongly Gevrey…

We study the second cohomology group with coefficients in the adjoint module for a class of solvable Lie algebras $\mathcal{R}_{\mathcal{T}}$ that arise as maximal solvable extensions of nilpotent Lie algebras $\mathcal{N}$ of maximal rank.…

We study generalized solutions of an evolutionary equation related to some densely defined skew-symmetric operator in a real Hilbert space. We establish existence of a contractive semigroup, which provides generalized solutions, and suggest…

In this note we present a framework which allows to prove an abstract existence result for evolution equations with pseudo-monotone operators. The assumptions on the spaces and the operators can be easily verified in concrete examples.

In this work we obtain sufficient conditions for a variety with a torus action of complexity one to have a finite number of automorphism group orbits.

We construct quantum evolution operators on the space of states, that realize the metaplectic representation of the modular group SL(2,Z_2^n). This representation acts in a natural way on the coordinates of the non-commutative 2-torus and…

One possible way to fix partly a ``canonical definition'' of $\tau$-functions beyond the conventional KP/Toda framework could be to postulate that evolution operators are {\it always} group elements. We discuss implications of this…

In this paper we consider a class of $p$-evolution equations of arbitrary order with variable coefficients depending on time and space variables $(t,x)$. We prove necessary conditions on the decay rates of the coefficients for the…

We study the solvability of a class of fully nonlinear equations on the flat torus. The equations arise in the study of some Calabi-Yau type problems in torus bundles.

In this paper, we investigate the three dimensional stationary compressible Navier-Stokes equations, and obtain Liouville type theorems if a smooth solution $(\rho, \mathbf{u})$ satisfies some suitable conditions. In particular, our results…

This paper is concerned with unbounded observation operators for non-autonomous evolution equations. Fix $\tau > 0$ and let $\left(A(t)\right)_{t \in [0,\tau]} \subset \mathcal{L}(D,X)$, where $D$ and $X$ are two Banach spaces such that $D$…