On weak, strong and classical solutions of the Hopf equation. An example of F.D.E. of second order
Let (M,g) be a compact Riemannian manifold of \(\dim M=d\) with or without boundary \(\partial M\). We denote by \(\overset \circ X_{\sigma}(M)\) the space of solenoidal vector fields on M which vanish near the boundary. H stands for the completion of the above space with respect to \(L^ 2\)-norm, denoted by \(| \cdot |\). \(V^ s\) stands for the completion of \(\overset \circ X_{\sigma}(M)\) in the Sobolev space of order \(s\in Z\), whose norm is denoted by \(\| \cdot \|_ s\). For 1-forms, we introduce \(\overset \circ \Lambda^ 1_{\sigma}(M)\) analogously. The completions of it with corresponding norms are denoted by \(\tilde H\) and \(\tilde V^ s\), respectively. The space of symmetric tensor fields with 2 contravariant (or covariant) indices is denoted by \(ST_ 2(M)\) (or \(ST^ 2(M).)\) Our aim of this paper is to `solve' the following Functional Derivative Equation (F.D.E.): (I) Find a functional W(t,\(\eta\)), for \(t\in (0,\infty)\), \(\eta\in \overset \circ \Lambda^ 1_{\sigma}(M)\) satisfying \[ \frac{\partial}{\partial t}W(t,\eta)=\int_{M}[- i\{\frac{\partial}{\partial x^ j}\eta_ i(x)- \Gamma^{\ell}_{ij}(x)\eta_{\ell}(x)\}\frac{\delta^ 2W(t,\eta)}{\delta \eta_ i(x)\delta \eta_ j(x)}+ \] \[ +\nu (\Delta \eta)_ i(x)\frac{\delta W(t,\eta)}{\delta \eta_ i(x)}+i\eta_ j(x)f^ j(c,t)W(t,\eta)]d_ gx, \] \[ \frac{1}{\sqrt{g(x)}}\frac{\partial}{\partial x^ i}\{\sqrt{g(x)}\frac{\delta W(t,\eta)}{\delta \eta_ i(x)}\}=0,\quad W(t,0)=1\quad W(0,\eta)=W_ 0(\eta). \] Here \(\eta (x)=\eta_ j(x)dx^ j\in \overset \circ \Lambda^ 1_{\sigma}(M)\), and \(f(x,t)=f^ j(x,t)(\partial /\partial x^ j)\in \overset \circ X_{\sigma}(M)\) for a.e. t, \(W_ 0(\eta)\) is a given positive definite functional on \(\overset \circ \Lambda^ 1_{\sigma}(M)\) satisfying \[ W_ 0(0)=1\text{ and } \frac{1}{\sqrt{g(x)}}\frac{\partial}{\partial x^ j}\{\sqrt{g(x)}\frac{\delta W_ 0(\eta)}{\delta \eta_ j(x)}\}=0. \]
scientific article; zbMATH DE number 4163374