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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)].

##### MSC:
 35K55 Nonlinear parabolic equations 35B65 Smoothness and regularity of solutions to PDEs
##### Keywords:
periodic boundary conditions; Galerkin method
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