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Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. (English) Zbl 0803.35068

From the introduction: We consider the Cauchy problem for semilinear heat equations on \(\mathbb{R}^ n\) of the following form: \[ (1) \quad {\partial u \over \partial t} (t,x) = \Delta u(t,x) + f \bigl( u(t,x) \bigr) \quad \text{in} \quad ]0,\infty [\times \mathbb{R}^ n, \qquad (2) \quad u(0,x) = a(x) \quad \text{on} \quad \mathbb{R}^ n, \] where \(f(\sigma)\) is a locally Lipschitz continuous function on \(\mathbb{C}\) satisfying (3) \(\bigl | f (\sigma) - f (\rho) \bigr | \leq C | \sigma - \rho | \bigl( 1+ | \sigma |^{\gamma - 1} + | \rho |^{\gamma - 1} \bigr)\), or, more strongly, (4) \(f(0) = 0\) and \(| f (\sigma) - f(\rho) | \leq C | \sigma - \rho | (| \sigma |^{\gamma - 1} + | \rho |^{\gamma - 1})\) for every \(\sigma\), \(\rho \in \mathbb{C}\) with some constants \(\gamma>1\) and \(C>0\).
We also consider the Cauchy problem for the Navier-Stokes equation on \(\mathbb{R}^ n\) for \(n \geq 2\) of the following form: \[ {\partial u \over \partial t} - \Delta_ xu + (u \cdot \nabla_ x) u + \nabla_ x \pi = 0 \quad \text{in} \quad ]0, \infty [\times \mathbb{R}^ n, \tag{5} \]
\[ (6) \qquad \nabla_ x \cdot u = 0 \quad \text{in} \quad ]0, \infty [\times \mathbb{R}^ n, \qquad (7) \qquad u(0,x) = a(x) \quad \text{on} \quad \mathbb{R}^ n. \] The purpose of this paper is to construct new function spaces in the same way as the Besov spaces, based on the Morrey spaces in place of the standard \(L^ p\)-spaces, and to show that, if the initial data \(a(x)\) belongs to some function spaces above and its norm is sufficiently small, then the Cauchy problems (1)–(2) and (5)–(7) admit unique time-global strong solutions with a bound near \(t=0\), provided that the function \(f(\sigma)\) in (1) satisfies (4) with some constant \(\gamma > 1 + 2/n\). We also introduce a local version of the above function spaces, and show that the above Cauchy problems, with \(f(\sigma)\) satisfying (3) in (1), admit unique time-local strong solutions with a bound near \(t=0\) for initial data in the local function spaces under an additional assumption.

MSC:

35K55 Nonlinear parabolic equations
35Q30 Navier-Stokes equations
35K15 Initial value problems for second-order parabolic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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