Journal of Evolution Equations最新文献

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).

In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian (mathcal {L}) on the Heisenberg group (mathbb {H}^n) with power type nonlinearity (|u|^p) and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space (dot{H}^{-gamma }_{mathcal {L}}(mathbb {H}^n), gamma >0), on (mathbb {H}^n). In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent (p_{text {crit}}(Q, gamma )=1+frac{4}{Q+2gamma },) for (gamma in (0, frac{Q}{2})), where (Q:=2n+2) is the homogeneous dimension of (mathbb {H}^n). More precisely, we establish

• A global-in-time existence of small data Sobolev solutions of lower regularity for (p>p_{text {crit}}(Q, gamma )) in the energy evolution space;

• A finite time blow-up of weak solutions for (1<p<p_{text {crit}}(Q, gamma )) under certain conditions on the initial data by using the test function method.

Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.

We study the solvability of the second boundary value problem of the Lagrangian mean curvature equation arising from special Lagrangian geometry. By the parabolic method, we obtain the existence and uniqueness of the smooth uniformly convex solution, which generalizes the Brendle–Warren’s theorem about minimal Lagrangian diffeomorphism in Euclidean metric space.

In this article, we prove that small localized data yield solutions to Kawahara-type equations which have linear dispersive decay on a finite time scale depending on the size of the initial data. We use the similar method used by Ifrim and Tataru to derive the dispersive decay bound of the solutions to the KdV equation, with some steps being simpler. This result is expected to be the first result of the small data global bounds of the fifth-order dispersive equations with quadratic nonlinearity.