Journal of Evolution Equations最新文献

We consider differential operators A that can be represented by means of a so-called closure relation in terms of a simpler operator (A_{{text {ext}}}) defined on a larger space. We analyse how the spectral properties of A and (A_{{text {ext}}}) are related and give sufficient conditions for exponential stability of the semigroup generated by A in terms of the semigroup generated by (A_{{text {ext}}}). As applications we study the long-term behaviour of a coupled wave–heat system on an interval, parabolic equations on bounded domains that are coupled by matrix-valued potentials, and of linear infinite-dimensional port-Hamiltonian systems with dissipation on an interval.

In this paper, we consider the inhomogeneous nonlinear Schrödinger equation (ipartial _t u +Delta u =K(x)|u|^alpha u,; u(0)=u_0in H^1({mathbb {R}}^N),; Nge 3,; |K(x)|+|x||nabla K(x)|lesssim |x|^{-b},; 0<b< min (2, N-2),; 0<alpha <{(4-2b)/(N-2)}). We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted (L^2)-space for a new range (alpha _0(b)<alpha <(4-2b)/N). The value (alpha _0(b)) is the positive root of (Nalpha ^2+(N-2+2b)alpha -4+2b=0,) which extends the Strauss exponent known for (b=0). Our results improve the known ones for (K(x)=mu |x|^{-b}), (mu in {mathbb {C}}). For general potentials, we highlight the impact of the behavior at the origin and infinity on the allowed range of (alpha ). In the defocusing case, we prove decay estimates provided that the potential satisfies some rigidity-type condition which leads to a scattering result. We give also a new scattering criterion taking into account the potential K.

This paper studies regularity properties of the weak solutions to the 3D Navier–Stokes equations with damping in the whole space and bounded domains. We find the space restriction on the initial velocity to guarantee the local existence of strong solutions. Based on it, we complete the existence results for the global strong solutions in the whole space and improve the restriction on the damping exponent for the existence of the global strong solutions in the bounded domains.

We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative (L^2)-gradient flow of the elastic energy. For any initial curve lying merely in the energy space we show existence and parabolic smoothing of the solution. Applying previous results on long-time existence and proving a constrained Łojasiewicz–Simon gradient inequality we furthermore show convergence to a critical point as time tends to infinity.

An algebraic upper bound for the decay rate of solutions to the Navier–Stokes and Navier–Stokes–Coriolis equations in the critical space (dot{H} ^{frac{1}{2}} (mathbb {R}^3)) is derived using the Fourier splitting method. Estimates are framed in terms of the decay character of initial data, leading to solutions with algebraic decay and showing in detail the roles played by the linear and nonlinear parts. The proof is carried on purely in the critical space, as no (L^2 (mathbb {R}^3)) estimates are available for the solution. This is the first instance in which such a method is used for obtaining decay bounds in a critical space for a nonlinear equation.

In this paper, we prove an interpolation inequality on Riemann–Liouville fractional integrals and then use it to study the strong stability and semi-uniform stability of fractional resolvent families of order (0<alpha <2). Let A denote the generator of a bounded fractional resolvent family. We show that if (sigma (A)cap (textrm{i}{mathbb {R}})^alpha ) is countable and (sigma _r(A) cap (textrm{i}{mathbb {R}})^alpha =varnothing ), then the bounded fractional resolvent family is strongly stable. And the semi-uniform stability of the fractional resolvent family is equivalent to (sigma (A)cap (textrm{i}{mathbb {R}})^alpha =varnothing ). Moreover, the relation between decay rates of semi-uniform stability and growth of the resolvent of A along ((textrm{i}{mathbb {R}})^alpha ) is given.

The aim of this paper is to investigate the stability of a stationary solution of free boundary problems of the incompressible Navier–Stokes equations in a three-dimensional bounded domain with surface tension. More precisely, this article proves that if the initial angular momentum is sufficiently small and if the initial configuration is sufficiently close to equilibrium, then there exists a global classical solution that converges exponentially fast to a uniform rigid rotation of the liquid as (t rightarrow infty ) with respect to a certain axis. The proof of the unique existence of a stationary solution is also given.

In this paper, we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy–Leray-type potential. More precisely, we consider the problem

where (N> 2s), (0<s<1) and (0<lambda <Lambda _{N,s}), the optimal constant in the fractional Hardy–Leray inequality. In particular, we show the existence of a critical existence exponent (p_{+}(lambda , s)) and of a Fujita-type exponent (F(lambda ,s)) such that the following holds:

• Let (p>p_+(lambda ,s)). Then there are not any non-negative supersolutions.

• Let (p<p_+(lambda ,s)). Then there exist local solutions, while concerning global solutions we need to distinguish two cases:

  • Let ( 1< ple F(lambda ,s)). Here we show that a weighted norm of any positive solution blows up in finite time.

  • Let (F(lambda ,s)<p<p_+(lambda ,s)). Here we prove the existence of global solutions under suitable hypotheses.

In this paper, we are concerned with (L^2) decay of weak solutions of several well-known incompressible fluid models, such as the n-dimensional ((nge 2)) Navier–Stokes equations with fractional hyperviscosity, the three-dimensional convective Brinkman–Forchheimer equations and the generalized SQG equation. A new approach, different from the classical Fourier splitting method develpoed by Schonbek (Commun Partial Differ Equ 11:733–763, 1986) and the spectral representation technique by Kajikiya and Miyakawa (Math Z 192:135-148,1986), is presented. By using the new approach, we can recover and improve some known decay results.

We consider the setting of manifolds with ends which are obtained by compact perturbation (gluing) of ends of the form ({mathbb {R}}^{n_i}times {mathcal {M}}_i). We investigate the family of vertical resolvent ({sqrt{t}nabla (1+tDelta )^{-m}}_{t>0}), where (mge 1). We show that the family is uniformly continuous on all (L^p) for (1le ~p~le ~min _{i}n_i). Interestingly, this is a closed-end condition in the considered setting. We prove that the corresponding maximal function is bounded in the same range except that it is only weak-type (1, 1) for (p=1). The Fefferman-Stein vector-valued maximal function is again of weak-type (1, 1) but bounded if and only if (1<p<min _{i}n_i), and not at (p=min _{i}n_i).