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Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations. (English) Zbl 0940.35119
Parabolic problems of the type \[ \begin{aligned} & \dot x= Ax+ f(t,x),\quad t> t_0,\\ & x(t_0)= x_0,\end{aligned}\tag{1} \] where the linear operator \(A: D(A)\subset X^0\to X^0\) is such that \(-A\) is a sectorial operator in the Banach space \(X^0\), is considered. A local existence and uniqueness theorem for this problem is proved when the nonlinearity \(f\) satisfies certain critical conditions. By \(X^\alpha\), \(\alpha\geq 0\) the fractional power spaces associated to the operator \(A\), and by \(e^{At}\) the analytic semigroup generated by \(A\) are denoted. There are used the following definitions:
Definition 1. We say that \(x: [t_0, \tau]\to X^1\) is an \(\varepsilon\)-regular mild solution to (1) if \(x\in \mathbb{C}([t_0,\tau], X^1)\cap ((t_0, \tau], X^{1+\varepsilon})\), and \(x(t)\) satisfies \[ x(t)= e^{A(t- t_0)} x_0+ \int^t_{t_0} e^{A(t- s)}f(s, x(s)) ds. \] Definition 2. For \(\varepsilon\geq 0\), we will say that a map \(g\) is an \(\varepsilon\)-regular map relative to the pair \((X^1,X^0)\) if there exist \(\varrho> 1\), \(\gamma(\varepsilon)\) with \(\varrho\varepsilon\leq \gamma(\varepsilon)< 1\), and a constant \(c\), such that \(g: X^{1+\varepsilon}\to X^{\gamma(\varepsilon)}\) and \[ \|g(x)- g(y)\|_{X^{\gamma(\varepsilon)}}\leq c\|x- y\|_{X^{1+ \varepsilon}}(\|x\|^{\varrho- 1}_{X^{1+ \varepsilon}}+ \|y\|^{\varrho- 1}_{X^{1+ \varepsilon}}+ 1),\qquad \forall x,y\in X^{1+ \varepsilon}. \] The main results of this paper says that if \(f(t,\cdot)\) is an \(\varepsilon\)-regular map relative to the pair \((X^1,X^0)\) for some \(\varepsilon> 0\), then we will have existence and uniqueness of \(\varepsilon\)-regular mild solutions for problem (1). As examples, the authors chose to study the Navier-Stokes equations in the Hilbert setting, and the heat equations in the \(L^1\) and \(W^{1,q}\) setting.

MSC:
35K90 Abstract parabolic equations
35Q30 Navier-Stokes equations
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