## On the boundedness and decay of moments of solutions to the Navier-Stokes equations.(English)Zbl 1027.35095

The authors consider the Navier-Stokes equations (NSE) on $$\mathbb{R}^n$$, $$2\leq n\leq 5$$ and prove estimates of the type $\int_{\mathbb{R}^n}|x |^k|u|^2 dx\leq C(1+t)^{-2\mu(1-k/n)}, \tag{1}$ with $$k\leq n$$, $$\mu \geq{1\over 2}$$, for solutions $$u$$ that satisfy $$\|u\|_2\leq C(1+t)^{-\mu}$$, $$t\geq 0$$. To this end they consider the linear equation $v_t-\Delta v+(u' \nabla)v+ \nabla P(u',v)=0,\;u'(0)=u_0, \tag{2}$ where $$P(\cdot,\cdot)$$ is a bilinear form defined in terms of the Riesz transform and where the given $$u'$$ satisfies $u'\in C\bigl([0,T],\;L^2(\mathbb{R}^n)^n\bigr) \cap L^2\bigl([0,T], H^1(\mathbb{R}^n)^n\bigr). \tag{3}$ Given $$u_0\in L^2(\mathbb{R}^n)^n\cap L^r(\mathbb{R}^n)^n$$, $$r\geq 4$$, (2), (3) defines a sequence $$u_k,k\geq 0$$, where $$v=u_{k+1}$$, $$u'=u_k$$ in (2), (3). An element $$u$$ subject to (3) is an admissible solution on $$[0,T]$$ with initial data $$u_0$$ if the sequence $$u_k$$, $$k\geq 0$$ generated by (2), (3) converges to $$u$$ in $$L^2([0,T]$$, $$L^2(\mathbb{R}^n)^n)$$. Theorem 2.4 states that if $$u_0\in L^2(\mathbb{R}^n)^n\cap L^r(\mathbb{R}^n)^n$$ and $$\text{div} u_0=0$$, there is a $$T>0$$ and an $$u$$ with $u\in C\bigl([0,T], L^2(\mathbb{R}^n)^n \cap L^r(\mathbb{R}^n)^n \bigr)\cap L^2\bigl([0,T],\;H^1(\mathbb{R}^n)^n \bigr)\tag{4}$ which is an admissible solution on $$[0,T]$$ and that $$u$$ is a weak solution of NSE which satisfies an energy inequality. Decay properties such as (1) are now proved for admissible solutions. In fact theorems are obtained which state that if $$\|u_0\|_r$$ is small, then the associated admissible solution $$u$$ is global and satisfies estimates of type (1). Similar results hold in case where $$u$$ is not necessarily global. The proofs are based on rather involved estimates.

### MSC:

 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 35A35 Theoretical approximation in context of PDEs 35B41 Attractors 76D05 Navier-Stokes equations for incompressible viscous fluids