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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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##### References:
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