Analyticity of solutions to fully nonlinear parabolic evolution equations on symmetric spaces
The authors consider the following initial value problem: \(\{{{du}\over{dt}}+P(u)=0, u(0)=u_0\}\), where \(u\in C([0,t^+),B)\cap C^1([0,t^+),E_0)\), where \(t^+\equiv t^+(u_0)>0\) stands for the maximal existence time of \(u\), \(E_0\) is a Banach space, \(B\subset E_1\) is open in the Banach space \(E_1\) and \(P\in C^\omega(B,E_0)\). It is assumed that \(E_1\) is continuously injected and dense in \(E_0\). Furthermore, it is assumed that \(\forall u\in{\mathcal L}(E_1,E_0)\) posses the property of maximal regularity in the sense of Da Prato-Grisvard. In particular the spaces \(E_j\), \(j=0,1\), are built by means of numerical functions, of a suitable class, on \([0,t^+)\times M\), with \(M\) a closed symmetric Riemannian manifold. Then, it is proven that if \(P\) is compatible with the underlying Lie structure, above system has real analytic solutions \(u\in C^\omega([0,t^+)\times M)\). The authors emphasize that this theorem is a special case of another more general one proved in the same paper. Some applications of this general result are also considered. More precisely, Bellman's equation and a class of mean curvature flows are studied in this framework. Reviewer's remark: The above results can be directly gained in the framework of the geometric theory of PDEs. In fact, we can more generally consider the following differential equation: \(u_t+P(u)=0\), on the fiber bundle \(\pi:W\equiv{\mathbb R}\times M\times{\mathbb R}\to {\mathbb R}\times M\), \((t,x^\alpha,u)\mapsto(t,x^\alpha)\), \(1\leq\alpha\leq n\), where \(M\) is a smooth \(n\)-dimensional manifold. Let us assume, for simplicity, that \(P\) is defined at the zero order, i.e., does not depend on derivatives of \(u\). Then we get a first order differential equation \(E_1\subset J{\mathcal D}^1(W)\), locally defined by the function \(F\equiv u_t+P(u)\) on \(J{\mathcal D}^1(W)\). One can prove that this equation is involutive formally integrable, therefore in the analytic case it is also completely integrable. In fact, \(\dim E_1=2n+2+(\dim ((g_1)_{+1})_q={{(n+1)n}\over{2}}= \dim(E_1)_{+1}={{n^2+5n+4}\over{2}}\). Here \(((g_1)_{+1})_q\) is the symbol of the first prolongation \((E_1)_{+1}\) of \(E_1\) and \(q\in E_1\). This assures the surjectivity \((E_1)_{+1}\to E_1\). Furthermore, one has also \((\dim ((g_1)_{+1})_q={{(n+1)n}\over{2}}=\sum_{0\leq i\leq n}(\dim ((g_1)^{(i)}))_q=n+(n-1)+\cdots+n-(n-1)\). Here \(((g_1)^{(i)})_q=\zeta_1\rfloor(\cdots\zeta_i\rfloor((g_1)^{(i)})_q)\cdots)\), with \(\{\zeta_i\}_{1\geq i\leq n+1}\) a local natural basis for the tangent space \(T_p({\mathbb R}\times M)\). Note that the Cartan distribution, \({\mathbf E}_1\subset TE_1\), is of dimension \((2n+1)\), generated by the vector fields \(\{\xi\equiv\partial t-P(u)(\partial u-P_u\partial u_t);\zeta_\alpha\equiv -u_\alpha P_u\partial u_t;\chi_\alpha\equiv \partial u_\alpha\}\) on \(E_1\). Therefore a regular solution of \(E_1\) can be identified with an \((n+1)\)-dimensional smooth (or analytic) submanifold of \(E_1\) tangent to \({\mathbf E}_1\), that can be diffeomorphically projected on \({\mathbb R}\times M\), via the canonical projection \(\pi_1:J{\mathcal D}^1(W)\to {\mathbb R}\times M\). In this framework Cauchy problems and boundary value problems can be solved also. Furthermore, by calculating the integral bordism group \(\Omega_n^{E_1}\), the properties of the global solutions can be described. Generalizations of such an equation, by considering more general operators \(P\), can be done similarly. Finally generalizations of such an approach can be obtained also in noncommutative manifolds.
scientific article; zbMATH DE number 2101651