Related papers: On exponential splitting methods for semilinear ab…
This paper investigates the two-dimensional stochastic steady-state Navier-Stokes(NS) equations with additive random noise. We introduce an innovative splitting method that decomposes the stochastic NS equations into a deterministic NS…
Prior to the recent development of symplectic integrators, the time-stepping operator $\e^{h(A+B)}$ was routinely decomposed into a sum of products of $\e^{h A}$ and $\e^{hB}$ in the study of hyperbolic partial differential equations. In…
The Computation of discrete Contractive semigroups becomes necessary when we deal with several types of evolution equations in Discretizable Hilbert spaces, in this work we study some properties of the discrete forms of the contractive…
Using the example of such a complicated problem as the Cauchy problem for the Navier-Stokes equation, we show how the Poincar\'e-Riemann-Hilbert boundary value problem enables us to construct effective estimates of solutions for this case.…
We investigate a smoothing property for strongly-continuous operator semigroups, akin to ultracontractivity in parabolic evolution equations. Specifically, we establish the stability of this property under certain relatively bounded…
We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schr\"odinger (NNLS) equation $ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with the step-like initial data close to the ``shifted step function''…
Let $\mathcal{X}$ be a real separable Hilbert space. Let $C$ be a linear, bounded and positive operator on $\mathcal{X}$ and let $A$ be the infinitesimal generator of a strongly continuous semigroup on $\mathcal{X}$. Let $\{W(t)\}_{t\geq…
A new type of low-regularity integrator is proposed for Navier-Stokes equations, coupled with a stabilized finite element method in space. Unlike the other low-regularity integrators for nonlinear dispersive equations, which are all fully…
We present a spectral mapping theorem for continuous semigroups of operators on any Banach space $E$. The condition for the hyperbolicity of a semigroup on $E$ is given in terms of the generator of an evolutionary semigroup acting in the…
The algebraic structures underlying quasi-exact solvability for spin 1/2 Hamiltonians in one dimension are studied in detail. Necessary and sufficient conditions for a matrix second-order differential operator preserving a space of wave…
We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R^n. We prove well-posedness in…
We consider a time-fractional semilinear parabolic abstract Cauchy problem for a time-dependent sectorial operator $A(t)$ which satisfies the Acquistapace-Terreni conditions. We first prove local existence results for the mild solution of…
The obstacle problem is a class of free boundary problems which finds applications in many disciplines such as porous media, financial mathematics and optimal control. In this paper, we propose two operator-splitting methods to solve the…
The aim of this paper is twofold: (1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of…
In $L_2(\mathbb{R}^d;{\mathbb C}^n)$ we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}_\varepsilon$ with periodic coefficients depending on ${\mathbf x}/\varepsilon$. We study the behavior of the…
Coupled multi-physics problems are encountered in countless applications and pose significant numerical challenges. Although monolithic approaches offer possibly the best solution strategy, they often require ad-hoc preconditioners and…
We study two seminal approaches, developed by B. Simon and J. Kisy\'nski, to the well-posedness of the Schr\"odinger equation with a time-dependent Hamiltonian. In both cases the Hamiltonian is assumed to be semibounded from below and to…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
We investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove existence results by applying Schauder's fixed point technique. Moreover, we show fundamental…
We consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the existence of solutions…