Sentence examples for smooth solution on from inspiring English sources

Let ((u, n, p)) be a local smooth solution on ([0,tilde{T})) for some (tilde{T}leqinfty).

Under the assumptions of Theorem 1.1, the Cauchy problem (2.1), (2.3) admits a unique global smooth solution on ([0,infty) times mathbb{R}^{m}). Noting (1.4), by the maximum principle we obtain the result that, on the existence domain of smooth solution, we have r t,theta)> 0. (2.46).

If (fin C^{1}), (x_{0}in L^{infty }) and (vert x_{0}(theta_{1},ldots,theta_{n})vert >0), then the Cauchy problem (1.2), (1.3) admits a unique global smooth solution on ([0,infty) times mathbb{R}^{n}). The paper is organized as follows.

If (fin C^{1}) and (r_{0}in L^{infty }), then the Cauchy problem (2.5) admits a unique global smooth solution on ([0,infty times mathbb{R}^{m}). Now we turn to a consideration of the Cauchy problem (2.1) (i.e., (2.1a - 2.1b 2.1a - 2.1be have the following.

For λ ≥ 0, the network-constrained logistic regression model is presented as (3) L λ, β = − ∑ i = 1 n y i log ⁡ f X ′ i β + 1 − y i log ⁡ 1 − f X ′ i β + λ β T L β, where the first term in (3) is the log-likelihood function of the logistic model and the second term is a network constraint based on the Laplacian matrix, which induces a smooth solution of β on the graph.

Let ((rho, mathrm {u}, mathrm {H})) be a smooth solution of (1.1 - 1.6 1.1 - 1.6atimes(0,infty)).

This is how the network-constrained coefficient β T Lp β induces a smooth solution of β on the known network.

Let w be a smooth solution of (L_{epsilon, A}w=f) on Q.

A smooth solution of the 2D Euler equation on a bounded domain exists and is unique in a natural class locally in time, but it blows up in finite time in the sense of its vorticity losing continuity [6].

For global RBFs, spectral spatial convergence is observed for smooth solutions on quasi-uniform nodes, while high-order accuracy is observed for the local RBF stencil and partition of unity approaches.

In 1979, Aronson and Bénilan obtained a celebrated second-order differential inequality of the form [2] sum_{i}frac{partial}{partial x^{i}} biggl(pu^{p-2} frac{partial u}{partial x^{i}} biggr geq-frac{k}{t},quad k:=frac{n}{n(p-1)+2}, which applies to all positive smooth solutions of (1.1) defined on the whole Euclidean space on the condition that (p>1-frac{2}{n}).