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Gevrey regularity for nonlinear analytic parabolic equations. (English) Zbl 0907.35061
The authors consider the scalar equation \[ u_t- \nu\Delta u+G(u, \nabla u)= 0 \tag{*} \] with initial condition \(u(x,0)=u_0\) in a cube of \(\mathbb{R}^n\), with periodic boundary conditions. Denoting \((1-\Delta)\) by \(A\) and \(F(u,\nabla u) =G(u, \nabla u)-u\), the initial problem (*) is equivalent to \[ u_t+Au+ F(u,\nabla u) =0, \quad u(x,0)= u_0. \tag{**} \] Let us denote by \(H^s_{\text{per}}\) the usual \(L^2\) Sobolev space with periodic boundary conditions, and by \(G^{p/2}_\sigma (\Omega) =D(A^{p/2} e^{\sigma A^{1/2}})\) where \(\Omega= [0,l]^n\). This space is a Hilbert space with respect to the scalar product \((v,w)_{G^{p/2}_\sigma (\Omega)} = \sum_{j \in\mathbb{Z}^n} v_j\overline w_j (1+| j|^2)^p e^{2\sigma (1+ | j|^2)^{1/2}}\) where \(v_j\) and \(w_j\) are the Fourier coefficients of \(v\) and \(w\), respectively. The authors call \(u\in {\mathcal C} ([0,T]_j H^p_{\text{per}} (\Omega)) \cap L^2([0,T]\), \(D(t))\) a regular solution of (**) if \({du\over dt} \in L^2 ([0,T]\), \(L^2_{\text{per}} (\Omega))\) and \[ \left( {du(t) \over dt}, \varphi \right)_{L^2} +\bigl(A^{1/2} u(t),A^{1/2} \varphi \bigr)_{L^2} + \biggl(F \bigl(u(t, \nabla u(t) \bigr), \varphi \biggr)_{L^2} =0 \] for every \(\varphi\in H^1_{\text{per}} (\Omega)\) a.e. on \([0,T]\). Concerning \(F\) it is supposed to be real analytic.
The main result of the paper states that if \(u_0\in H^p_{\text{per}} (\Omega)\) with \(p> {n\over 2}\) and \(| u_0 |_{H^p_{\text{per} } (\Omega)}\leq M_0\) then there exists a \(T^*= T^*(M_0, \nu,l)>0\) such that (**) has an unique solution \(u\) on \([0,T^*]\), with initial data \(u_0\), and \(u(\cdot,t)\in G_t^{p/2} (\Omega)\) for \(t\in [0,T^*]\). For the proof one uses that if \(p>{n\over 2}\), \(G_\sigma^{p/2} (\Omega)\) is a Banach algebra, and an useful estimate for \(F(u)\) in the \(G_\sigma^{p/2} (\Omega)\) norm supposing \(u \in G_\sigma^{p/2} (\Omega)\). The existence uses a standard Galerkin method; in fact, the regular solution is a classical solution where \(T^*\) is given explicitly. The case where \({n\over 2} +1\geq p> {n\over 2}\) necessitates a supplementary condition on the majorant \(g\) of \(F\) and one obtains a similar result of existence and uniqueness.
Finally, for the case \(p> {n\over 2} +1\), if the initial data are smoother, i.e., \(u_0 \in H^p_{\text{per}} (\Omega)\) with \(p> {n\over 2} +1\), a result of existence and uniqueness for (**) holds. The method used is inspired by the work of J. Q. Dian, E. S. Titi and P. Holmes [Nonlinearity 6, 915-933 (1993; Zbl 0808.35133)].

35K55 Nonlinear parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
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