Finite-time blow-up of the heat flow of harmonic maps from surfaces

Stable Blowup Dynamics for the 1‐Corotational Energy Critical Harmonic Heat Flow

“…The existence of blowup solutions has been proved in various different geometrical settings; see in particular Chang, Ding, and Ye [2], Coron and Ghidaglia [3], Qing and Tian [19], and Topping [26]. In this paper we shall restrict our study to flows with symmetries that are better understood.…”

On the temporal decay of solutions to the two‐dimensional nematic liquid crystal flows

“…The nematic liquid crystal flow (1.1)-(1.4) is a simplified version of the Ericksen-Leslie model [3], [11], but it still retains most of the interesting mathematical properties. Mathematically, system (1.1)-(1.4) is a strongly coupled system between the incompressible Navier-Stokes (NS) equations (the case d ≡ d 0 (d 0 is a constant vector in S 1 ), e.g., [9], [10]) and the transported heat flows of harmonic map (the case u ≡ 0, see e.g., [1], [2], [18], [27], [28]), and thus, its mathematical analysis is full of challenges.…”

“…In the papers of Lin, Lin and Wang [13] and Hong [5], the authors proved that there exist global Leray-Hopf type weak solutions to (1.1)-(1.4) with suitable boundary condition in dimension two, and established that the solutions are smooth away from at most finitely many singular times which is similar as that for the heat flows of harmonic maps (cf. [1], [2], [27]). When the space dimension is three, Lin and Wang [17] established the existence of global weak solutions very recently when the initial orientation d 0 maps to the up-hemisphere S 2 + .…”

“…| u(t)| 2 + ∇d(t) 2 dξ. 1 For the reader's convenience, we show that the first integral is finite. Notice that 1 ≤ p < 2 implies that 1 − Notice that it is easy to see…”

Nash equilibria in N‐person games with super‐quadratic Hamiltonians

“…Essential for our analysis is the fact that H has a negative definite Jacobian H u in the case that |∇u| = 0. More precisely, it is well known that weak solutions may show finite time blow up behaviour if H u is positive definite; see, for example, [13] for the quadratic case. In the case that H u = 0 and q > 2 a loss of boundary conditions may occur; cf.…”