Journal of Evolution Equations最新文献

In an infinite-dimensional separable Hilbert space X, we study the realizations of Ornstein–Uhlenbeck evolution operators (P_{s,t}) in the spaces (L^p(X,gamma _t)), ({gamma _t}_{tin mathbb {R}}) being a suitable evolution system of measures for (P_{s,t}). We prove hypercontractivity results, relying on suitable Log-Sobolev estimates. Among the examples, we consider the transition evolution operator associated with a non-autonomous stochastic parabolic PDE.

In this paper, we show a blowup criterion of solution for a parabolic equation with critical exponential source and arbitrary positive initial energy, which generalizes the blowup conclusions in reference (Ishiwata et al. in J Evol Equ 21:1677–1716, 2021) for subcritical and critical initial energy cases that depend on the depth of the potential well. Additionally, the continuous dependence of the local solution on the initial data is proved in detail.

We study a free boundary value problem modelling the motion of a piston in a viscous compressible fluid. The fluid is modelled by 1D compressible Navier–Stokes equations with possibly degenerate viscosity coefficient, and the motion of the piston is described by Newton’s second law. We show that the initial boundary value problem has a unique global in time solution, and we also determine the large time behaviour of the system. Finally, we show how our methodology may be adapted to the motion of several pistons.

In this paper, we focus on investigating the existence of mild solutions and asymptotically almost periodic mild solutions for a class of partial differential inclusions. These inclusions involve a forcing multivalued function that relies on implicit spatial derivatives of the state variable. We introduce a novel approach to simplify the complexities associated with singularities when taking the (alpha )-norm.

Motivated by the competition model of two species with nonautonomous diffusion, we consider fully nonautonomous parabolic evolution equation of the form (frac{textrm{d}u}{textrm{d}t} + A(t)u(t) = f(t,u)+g(t)) in which the time-dependent family of linear partial differential operator A(t), the nonlinear term f(t, u), and the external force g is 1-periodic with respect to t. We prove the existence and uniqueness of a periodic solution of the above equation and study the inertial manifold for the solutions nearby that solution. We prove the existence of such an inertial manifold in the cases that the family of linear partial differential operators ((A(t))_{tin mathbb {R}}) generates an evolution family ((U(t,s))_{tge s}) satisfying certain dichotomy estimates, and the nonlinear term f(t, x) satisfies the (varphi )-Lipschitz condition, i.e., (left| f(t,x_1)-f(t,x_2)right| leqslant varphi (t)left| A(t)^{theta } (x_1-x_2)right| ) where (varphi (cdot )) belongs to some admissible function space on the whole line. Then, we apply our abstract results to the above-mentioned competition model of two species with nonautonomous diffusion.

We examine the large-time behavior of axisymmetric solutions without swirl of the Navier–Stokes equation in ({mathbb {R}}^3). We construct higher-order asymptotic expansions for the corresponding vorticity. The appeal of this work lies in the simplicity of the applied techniques: Our approach is completely based on standard (L^2)-based entropy methods.

We discover new cancellations upon (H^{2}(mathbb {R}^{n}))-estimate of the Hall term, (n in {2,3}). Consequently, first, we derive a regularity criterion for the 3-dimensional Hall-magnetohydrodynamics system in terms of horizontal components of velocity and magnetic fields. Second, we are able to prove the global regularity of the (2frac{1}{2})-dimensional electron magnetohydrodynamics system with magnetic diffusion ((-Delta )^{frac{3}{2}} (b_{1}, b_{2}, 0) + (-Delta )^{alpha } (0, 0, b_{3})) for (alpha > frac{1}{2}) despite the fact that ((-Delta )^{frac{3}{2}}) is the critical diffusive strength. Lastly, we extend this result to the (2frac{1}{2})-dimensional Hall-magnetohydrodynamics system with (-Delta u) replaced by ((-Delta )^{alpha } (u_{1}, u_{2}, 0) -Delta (0, 0, u_{3})) for (alpha > frac{1}{2}). The sum of the derivatives in diffusion that our result requires is (11+ epsilon ) for any (epsilon > 0), while the sum for the classical (2frac{1}{2})-dimensional Hall-magnetohydrodynamics system is 12 considering (-Delta u) and (-Delta b), of which its global regularity issue remains an outstanding open problem.

We are concerned with the Cauchy problem to the three-dimensional full compressible Navier–Stokes equations with zero heat conductivity. Under the condition that the initial energy is small enough, global existence of strong solutions is established. Especially, the initial density is allowed to have large oscillations. The key to estimate the pointwise lower and upper bounds of the density lies in the handling of the energy conservation equation and the boundedness of the (L^r)–norm of the gradient of the pressure.

We prove that for bounded, divergence-free vector fields in (L^1_textrm{loc}((0,+infty );BV_textrm{loc}(mathbb {R}^d;mathbb {R}^d))), regularisation by convolution of the vector field selects a single solution of the transport equation for any locally integrable initial datum. We recall the vector field constructed by Depauw in (C R Math Acad Sci Paris 337:249–252, 2003), which lies in the above class of vector fields. We show that the transport equation along this vector field has at least two bounded weak solutions for any bounded initial datum.

We consider the barotropic Euler equations with pairwise attractive Riesz interactions and linear velocity damping in the periodic domain. We establish the global-in-time well-posedness theory for the system near an equilibrium state if the coefficient of the Riesz interaction term is small. We also analyze the large-time behavior of solutions showing the exponential rate of convergence toward the equilibrium state as time goes to infinity.