Let B be a two-dimensional ball with radius R. We continue to study the shape of the stable steady states to ut = Du∆u + f(u, ξ) in B× R+, τξt = 1 |B| ZZ B g(u, ξ)dxdy in R+, ∂νu = 0 on ∂B × R+, where f and g satisfy the following: fξ(u, ξ) < 0, gξ(u, ξ) < 0, and there is a function k(ξ) such that gu(u, ξ) = k(ξ)fξ(u, ξ). This system includes a special case of the Gierer-Meinhardt system and th...

where detξ denotes the determinant of ξ, and (C4) W (PξQ) = W (ξ) for all ξ ∈ M and all P,Q ∈ SO(3) for N = 3, with SO(3) := {Q ∈ M : QQ = QQ = I3 and detQ = 1}, where I3 denotes the identity matrix in M 3×3 and Q is the transposed matrix of Q. (In fact, (C4) is an additional condition which is not related to (2). However, it means that W is frame-indifferent, i.e., W (Pξ) = W (ξ) for all ξ ∈ M...

where detξ denotes the determinant of ξ, and (C4) W (PξQ) = W (ξ) for all ξ ∈ M and all P,Q ∈ SO(3) for N = 3, with SO(3) := {Q ∈ M : QQ = QQ = I3 and detQ = 1}, where I3 denotes the identity matrix in M 3×3 and Q is the transposed matrix of Q. (In fact, (C4) is an additional condition which is not related to (2). However, it means that W is frame-indifferent, i.e., W (Pξ) = W (ξ) for all ξ ∈ M...

In this article, the two-dimensional magneto-hydrodynamic (MHD) equations are considered with only magnetic diffusion. Here the magnetic diffusion is given by D a Fourier multiplier whose symbol m is given by m(ξ) = |ξ| log(e + |ξ|) . We prove that there exists an unique global solution in H(R) with s > 2 for these equations when β > 1. This result improves the previous works which require that...

where detξ denotes the determinant of ξ, and (C4) W (PξQ) = W (ξ) for all ξ ∈ M and all P,Q ∈ SO(3) for N = 3, with SO(3) := {Q ∈ M : QQ = QQ = I3 and detQ = 1}, where I3 denotes the identity matrix in M 3×3 and Q is the transposed matrix of Q. (In fact, (C4) is an additional condition which is not related to (2). However, it means that W is frame-indifferent, i.e., W (Pξ) = W (ξ) for all ξ ∈ M...

We study the scaling limit of a branching random walk in static environment dimension d=1,2 and show that it is given by super-Brownian motion white noise potential. In 1 we characterize as unique weak solution to stochastic PDE ∂tμ=(Δ+ξ)μ+ 2νμξ˜ for independent space ξ space-time ξ˜. 2 requires paracontrolled theory process described via martingale problem. both dimensions prove persistence th...

We consider the Yang-Mills problem for a quantum Heisenberg manifold, which is a C∗-algebra defined by the (strict) deformation quantization of the ordinary Heisenberg manifold, in the setting of non-commutative differential geometry following Connes and Rieffel [Co] [Co1]. 1. Preliminaries Classical Yang-Mills theory is concerned with the set of connections (i.e. gauge potentials) on a vector ...

This note is intended as a supplement and clarification to the proof of Theorem 3.3 of [1]; namely, it is consistent that b = ℵ 1 yet for every ultrafilter U on ω there is a ≤ * chain {f ξ : ξ ∈ ω 2 } such that {f ξ /U : ξ ∈ ω 2 } is cofinal in ω/U. The general outline of the the proof remains the same. In other words, a ground model is taken which satisfies 2 ℵ0 = ℵ 1 and in which there is a ♦...