Journal of Evolution Equations最新文献

Abstract

We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity (a(lambda x, lambda ^sigma xi ) = lambda ^{1+sigma } a(x,xi )) for (lambda > 0) where (sigma > 0) is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin–Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol.

In this paper, we are concerned with multi-scale distribution-dependent stochastic differential equations driven by fractional Brownian motion (with Hurst index (H>frac{1}{2})) and standard Brownian motion, simultaneously. Our aim is to establish a large deviation principle for the multi-scale distribution-dependent stochastic differential equations. This is done via the weak convergence approach and our proof is based heavily on the fractional calculus.

In this work, we consider parabolic equations of the form

where (varepsilon ) is a parameter in ([0,varepsilon _0)), and ({A_{varepsilon }(t), tin {mathbb {R}}}) is a family of uniformly sectorial operators. As (varepsilon rightarrow 0^{+}), we assume that the equation converges to

The time-dependence found on the linear operators (A_{varepsilon }(t)) implies that linear process is the central object to obtain solutions via variation of constants formula. Under suitable conditions on the family (A_{varepsilon }(t)) and on its convergence to (A_0(t)) when (varepsilon rightarrow 0^{+}), we obtain a Trotter-Kato type Approximation Theorem for the linear process (U_{varepsilon }(t,tau )) associated with (A_{varepsilon }(t)), estimating its convergence to the linear process (U_0(t,tau )) associated with (A_0(t)). Through the variation of constants formula and assuming that (F_{varepsilon }) converges to (F_0), we analyze how this linear process convergence is transferred to the solution of the semilinear equation. We illustrate the ideas in two examples. First a reaction-diffusion equation in a bounded smooth domain (Omega subset {mathbb {R}}^{3})

where (a_varepsilon ) converges to a function (a_0), (f_{varepsilon }) converges to (f_0). We apply the abstract theory in this example, obtaining convergence of the linear process and solution. As a consequence, we also obtain upper-semicontinuity of the family of pullback attractors associated with each problem. The second example is a nonautonomous strongly damped wave equation

where (Delta _D) is the Laplacian operator with Dirichlet boundary conditions in a domain (Omega ) and we analyze convergence of solution as we perturb the fractional powers of the associated linear operator.

Abstract

The purpose of this paper is to give an exact division for the well-posedness and ill-posedness (non-existence) of the Hunter–Saxton equation on the line. Since the force term is not bounded in the classical Besov spaces (even in the classical Sobolev spaces), a new mixed space (mathcal {B}) is constructed to overcome this difficulty. More precisely, if the initial data (u_0in mathcal {B}cap dot{H}^{1}(mathbb {R}),) the local well-posedness of the Cauchy problem for the Hunter–Saxton equation is established in this space. Contrariwise, if (u_0in mathcal {B}) but (u_0notin dot{H}^{1}(mathbb {R}),) the norm inflation and hence the ill-posedness is presented. It’s worth noting that this norm inflation occurs in the low frequency part, which exactly leads to a non-existence result. Moreover, the above result clarifies a corollary with physical significance such that all the smooth solutions in (L^{infty }(0,T;L^{infty }(mathbb {R}))) must have the (dot{H}^1) norm.

This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for (L^1) initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as (L^1) asymptotic convergence without the assumption of bi-K-invariance.

Abstract

We consider degenerate Kolmogorov–Fokker–Planck operators $$begin{aligned} mathcal {L}u&=sum _{i,j=1}^{m_{0}}a_{ij}(x,t)partial _{x_{i}x_{j}} ^{2}u+sum _{k,j=1}^{N}b_{jk}x_{k}partial _{x_{j}}u-partial _{t}u&equiv sum _{i,j=1}^{m_{0}}a_{ij}(x,t)partial _{x_{i}x_{j}}^{2}u+Yu end{aligned}$$ (with ((x,t)in mathbb {R}^{N+1}) and (1le m_{0}le N) ) such that the corresponding model operator having constant (a_{ij}) is hypoelliptic, translation invariant w.r.t. a Lie group operation in (mathbb {R} ^{N+1}) and 2-homogeneous w.r.t. a family of nonisotropic dilations. The matrix ((a_{ij})_{i,j=1}^{m_{0}}) is symmetric and uniformly positive on (mathbb {R}^{m_{0}}) . The coefficients (a_{ij}) are bounded and Dini continuous in space, and only bounded measurable in time. This means that, setting $$begin{aligned} mathrm {(i)}&,,S_{T}=mathbb {R}^{N}times left( -infty ,Tright) , mathrm {(ii)}&,,omega _{f,S_{T}}(r) = sup _{begin{array}{c} (x,t),(y,t)in S_{T} Vert x-yVert le r end{array}}vert f(x,t) -f(y,t)vert mathrm {(iii)}&,,Vert fVert _{mathcal {D}( S_{T}) } =int _{0}^{1} frac{omega _{f,S_{T}}(r) }{r}dr+Vert fVert _{L^{infty }left( S_{T}right) } end{aligned}$$ we require the finiteness of (Vert a_{ij}Vert _{mathcal {D}(S_{T})}) . We bound (omega _{u_{x_{i}x_{j}},S_{T}}) , (Vert u_{x_{i}x_{j}}Vert _{L^{infty }( S_{T}) }) ( (i,j=1,2,...,m_{0}) ), (omega _{Yu,S_{T}}) , (Vert YuVert _{L^{infty }( S_{T}) }) in terms of (omega _{mathcal {L}u,S_{T}}) , (Vert mathcal {L}uVert _{L^{infty }( S_{T}) }) and (Vert uVert _{L^{infty }left( S_{T}right) }) , getting a control on the uniform continuity in space of (u_{x_{i}x_{j}},Yu) if (mathcal {L}u) is bounded and Dini-continuous in space. Under the additional assumption that both the coefficients (a_{ij}) and (mathcal {L}u) are log-Dini continuous, meaning the finiteness of the quantity $$begin{aligned} int _{0}^{1}frac{omega _{f,S_{T}}left( rright) }{r}left| log rright| dr, end{aligned}$$ we prove that (u_{x_{i}x_{j}}) and Yu are Dini continuous; moreover, in this case, the derivatives (u_{x_{i}x_{j}}) are locally uniformly continuous in space and time.

We study local boundedness and Hölder continuity of a parabolic equation involving the fractional p-Laplacian of order s, with (0<s<1), (2le p < infty ), with a general right-hand side. We focus on obtaining precise Hölder continuity estimates. The proof is based on a perturbative argument using the already known Hölder continuity estimate for solutions to the equation with zero right-hand side.

We show that solutions of the complete Euler system of gas dynamics perturbed by a friction term converge to a solution of the porous medium equation in the high friction/long time limit. The result holds in the largest possible class of generalized solutions–the measure–valued solutions of the Euler system.

The generic alignment conjecture states that for almost every initial data on the torus solutions to the Cucker–Smale system with a strictly local communication align to the common mean velocity. In this note, we present a partial resolution of this conjecture using a statistical mechanics approach. First, the conjecture holds in full for the sticky particle model representing, formally, infinitely strong local communication. In the classical case, the conjecture is proved when N, the number of agents, is equal to 2. It follows from a more general result, stating that for a system of any size for almost every data at least two agents align. The analysis is extended to the open space (mathbb {R}^n) in the presence of confinement and potential interaction forces. In particular, it is shown that almost every non-oscillatory pair of solutions aligns and aggregates in the potential well.