Journal of Evolution Equations最新文献

Let (Omega subset mathbb {R}^d) be a bounded open connected set with Lipschitz boundary. Let (A^N) and (A^D) be the Stokes Neumann operator and Stokes Dirichlet operator on (Omega ), respectively. We study the associated Stokes version of the Dirichlet-to-Neumann operator and show a Krein formula which relates these three Stokes version operators. We also prove a Stokes version of the Friedlander inequalities, which relates the Dirichlet eigenvalues and the Neumann eigenvalues.

We study rates of decay for (C_0)-semigroups on Banach spaces under the assumption that the norm of the resolvent of the semigroup generator grows with (|s|^{beta }log (|s|)^b), (beta , b ge 0), as (|s|rightarrow infty ), and with (|s|^{-alpha }log (1/|s|)^a), (alpha , a ge 0), as (|s|rightarrow 0). Our results do not suppose that the semigroup is bounded. In particular, for (a=b=0), our results improve the rates involving Fourier types obtained by Rozendaal and Veraar (J Funct Anal 275(10):2845–2894, 2018).

Our interest in this research is to prove the decay, as time tends to infinity, of a unique global solution for the supercritical case of the modified quasi-gesotrophic equation (MQG)

in the non-homogenous Sobolev space (H^{1+gamma -2alpha }(mathbb {R}^2)), where (alpha in (0,frac{1}{2})) and (gamma in (1,2alpha +1)). To this end, we need consider that the initial data for this equation are small. More precisely, we assume that (Vert theta _0Vert _{H^{1+gamma -2alpha }}) is small enough in order to obtain a unique (theta in C([0,infty );H^{1+gamma -2alpha }(mathbb {R}^2))) that solves (MQG) and satisfies the following limit:

We treat the linear heat equation in a periodic waveguide (Pi subset {{mathbb {R}}}^d), with a regular enough boundary, by using the Floquet transform methods. Applying the Floquet transform ({{textsf{F}}}) to the equation yields a heat equation with mixed boundary conditions on the periodic cell (varpi ) of (Pi ), and we analyse the connection between the solutions of the two problems. The considerations involve a description of the spectral projections onto subspaces ({{mathcal {H}}}_S subset L^2(Pi )) corresponding certain spectral components. We also show that the translated Wannier functions form an orthonormal basis in ({{mathcal {H}}}_S).

We consider the nonlocal Cahn-Hilliard equation with constant mobility and singular potential in three dimensional bounded and smooth domains. This model describes phase separation in binary fluid mixtures. Given any global solution (whose existence and uniqueness are already known), we prove the so-called instantaneous and uniform separation property: any global solution with initial finite energy is globally confined (in the (L^infty ) metric) in the interval ([-1+delta ,1-delta ]) on the time interval ([tau ,infty )) for any (tau >0), where (delta ) only depends on the norms of the initial datum, (tau ) and the parameters of the system. We then exploit such result to improve the regularity of the global attractor for the dynamical system associated to the problem.

We present some results concerning the control of the Burgers equation. We analyze a bi-objective optimal control problem and then the hierarchical null controllability through a Stackelberg–Nash strategy, with one leader and two followers. The results may be viewed as an extension to this nonlinear setting of a previous analysis performed for linear and semilinear heat equations. They can also be regarded as a first step in the solution of control problems of this kind for the Navier–Stokes equations.

Abstract

We consider incompressible Navier–Stokes equations in a bounded 2D domain, complete with the so-called dynamic slip boundary conditions. Assuming that the data are regular, we show that weak solutions are strong. As an application, we provide an explicit upper bound of the fractal dimension of the global attractor in terms of the physical parameters. These estimates comply with analogous results in the case of Dirichlet boundary condition.

This paper is concerned with the uniqueness and energy conservation of weak solutions for Electron-MHD system. Under suitable assumptions, we first show that the Electron-MHD system has a unique weak solution. In addition, we show that weak solution conserves energy if (nabla times bin L^2(0, T; L^4({mathbb {R}}^d))(dge 2)) or ( nabla times b in L^{frac{4d+8}{d+4}}left( 0, T; L^{frac{4d+8}{d+4}}({mathbb {R}}^{d})right) (d=2, 3, 4)).

Abstract

The purpose of this paper is to revisit previous works of the author with Helffer and Sjöstrand (arXiv:1001.4171v1. 2010; Int Equ Op Theory 93(3):36, 2021) on the stability of semigroups which were proved in the Hilbert case by considering the Banach case at the light of a paper by Latushkin and Yurov (Discrete Contin Dyn Syst 33:5203–5216, 2013).

The quasilinear Keller–Segel system

endowed with homogeneous Neumann boundary conditions is considered in a bounded domain (Omega subset {mathbb {R}}^n), (n ge 3), with smooth boundary for sufficiently regular functions D and S satisfying (D>0) on ([0,infty )), (S>0) on ((0,infty )) and (S(0)=0). On the one hand, it is shown that if (frac{S}{D}) satisfies the subcritical growth condition

and (C>0), then for any sufficiently regular initial data there exists a global weak energy solution such that ({ mathrm{{ess}}} sup _{t>0} Vert u(t) Vert _{L^p(Omega )}<infty ) for some (p > frac{2n}{n+2}). On the other hand, if (frac{S}{D}) satisfies the supercritical growth condition

and (c>0), then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value (alpha = frac{2}{n}) for (n ge 3), without any additional assumption on the behavior of D(s) as (s rightarrow infty ), in particular without requiring any algebraic lower bound for D. When applied to the Keller–Segel system with volume-filling effect for probability distribution functions of the type (Q(s) = exp (-s^beta )), (s ge 0), for global solvability the exponent (beta = frac{n-2}{n}) is seen to be critical.