Space–time fractional diffusion on bounded domains

Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space–time fractional diffusion equations on bounded domains, as well as probabilistic representations of these solutions, which are useful for particle tracking codes.     

Proper holomorphic self-maps of smooth bounded Reinhardt domains in C~n

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain of D'Angelo finite type in Cn (n > 1) is an automorphism.     

On canonical metrics on Cartan–Hartogs domains

The Cartan–Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan–Hartogs domain \(\Omega ^{B^{d_0}}(\mu )\) endowed with the canonical metric \(g(\mu ),\) we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space \(\mathcal {H}_{\alpha }\) of square integrable holomorphic functions on \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) with the weight \(\exp \{-\alpha \varphi \}\) (where \(\varphi \) is a globally defined Kähler potential for \(g(\mu )\) ) for \(\alpha >0\) , and, furthermore, we give an explicit expression of the Rawnsley’s \(\varepsilon \) -function expansion for \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) .\) Secondly, using the explicit expression of the Rawnsley’s \(\varepsilon \) -function expansion, we show that the coefficient \(a_2\) of the Rawnsley’s \(\varepsilon \) -function expansion for the Cartan–Hartogs domain \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is constant on \(\Omega ^{B^{d_0}}(\mu )\) if and only if \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.     

Proper holomorphic self-maps of smooth bounded Reinhardt domains in C~2

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain in C~2 is an automorphism.     

Decay of solutions for 2D Navier–Stokes equations posed on Lipschitz and smooth bounded and unbounded domains

Initial–boundary value problems for 2D Navier–Stokes equations posed on bounded and unbounded rectangles as well as on bounded and unbounded smooth domains were considered. The existence and uniqueness of regular global solutions in bounded rectangles and bounded smooth domains as well as exponential decay of solutions on bounded and unbounded domains were established.     

On the smooth transfer conjecture of Jacquet–Rallis for n=3

We obtain a relative Shalika germ expansion of orbital integrals appeared in the relative trace formulae Jacquet?CRallis when n=3. This is the first example where there are infinitely many nilpotent orbits. As an application we can prove the smooth transfer conjecture of Jacquet?CRallis for n=3.     

On the Kashiwara–Vergne conjecture

Let G be a connected Lie group, with Lie algebra . In 1977, Duflo constructed a homomorphism of -modules , which restricts to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a conjecture on the Campbell-Hausdorff series, which (among other things) extends the Duflo theorem to germs of bi-invariant distributions on the Lie group G. The main results of the present paper are as follows. (1) Using a recent result of Torossian (2002), we establish the Kashiwara–Vergne conjecture for any Lie group G. (2) We give a reformulation of the Kashiwara–Vergne property in terms of Lie algebra cohomology. As a direct corollary, one obtains the algebra isomorphism , as well as a more general statement for distributions.     

On the Leray–Hopf extension condition for the steady-state Navier–Stokes problem in multiply-connected bounded domains

Employing the approach of Takeshita (Pacific J Math 157:151–158, 1993), we give an elementary proof of the invalidity of the Leray–Hopf Extension Condition for certain multiply connected bounded domains of \({\mathbb {R}}^{n}\) , \(n=2,3\) , whenever the flow through the different components of the boundary is non-zero. Our proof is alternative to and, to an extent, more direct than the recent one proposed by Heywood (J Math Fluid Mech 13:449–457, 2011).     

On the stability of the state 1 in the non-local Fisher–KPP equation in bounded domains

We consider the non-local Fisher–KPP equation on a bounded domain with Neumann boundary conditions. Thanks to a Lyapunov function, we prove that, under a general hypothesis on the kernel involved in the non-local term, the homogenous steady state 1 is globally asymptotically stable. This assumption happens to be linked to some conditions given in the literature, which ensure that travelling waves link 0 to 1.     

Explicit formula of holomorphic automorphism group on complex homogeneous bounded domains

We known that the maximal connected holomorphic automorphism group Aut (D)(0) is a semi-direct product of the triangle group T(D) and the maximal connected isotropic subgroup Iso(D)(0) of a fixed point in the complex homogeneous bounded domain D and any complex homogeneous bounded domain is holomorphic isomorphic to a normal Siegel domain D(VN,F). In this paper, we give the explicit formula of any holomorphic automorphism in T(D(VN, F)) and Iso(D(VN,F))(0), where G(0) is the unit connected component of the Lie group G.     

On the Mumford–Tate conjecture for 1-motives

We show that the statement analogous to the Mumford–Tate conjecture for Abelian varieties holds for 1-motives on unipotent parts. This is done by comparing the unipotent part of the associated Hodge group and the unipotent part of the image of the absolute Galois group with the unipotent part of the motivic fundamental group.     

Propagation of chaos for the Keller–Segel equation over bounded domains

In this paper we rigorously justify the propagation of chaos for the parabolic–elliptic Keller–Segel equation over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller–Segel equation for bounded weak solutions.