数理解析研究所講究録
第 1905 巻 2014 年 30-39

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DECAY CHARACTERIZATION FOR SOLUTIONS TO DISSIPATIVEEQUATIONS IN TERMS OF THE INITIAL DATUM
C\’ESAR J. NICHE AND MAR\’IA E. SCHONBEKABSTRACT. By examining the Fourier transform of the initial datum near the origin,
we define the decay character of the datum and provide a method to study the lower
and upper algebraic rates of decay of solutions to a wide class of dissipative syskem
of equations.

1. INTRODUCTION
We address the study of decay rates of solutions to nonlinear dissipative evolution
equations satisfying the energy inequality( $E$ )$\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{3}}|u(x, t)|^{2}dx\leq-C\int_{\mathbb{R}^{3}}|\nabla^{\alpha}u(x, t)|^{2}dx,$

where $\alpha\in(0,1$ ]. The characterization of the decay rates is given first for a class of linear
systems by introducing the concept of decay character, a number associated to the initial
datum that describes its behavior near the origin in frequency space. We then study
nonlinear systems with the underlying linear systems for which we have already obtained
decay rates. The decay character and the Fourier Splitting method are then used toobtain upper and lower bounds for decay of solutions to appropriate nonlinear dissipativeequations, both in the incompressible and compressible case. The method derived in thispaper can be applied to most of the equations that satisfy (E) . It works for systems like
Navier-Stokes, MHD, Quasi-Geostrophic equations and certain compressible systems.
-energy decay for weakWe recall the original question of Leray: how does thesolutions of the Navier-Stokes equations?. We would like to use the decay character inorder to give a concise answer to this question not only for the solutions to the NavierStokes equations, but for the class of all solutions to dissipative systems satisfying (E) .
Our goal is to, given the decay character of the initial datum, know whether the solution
with that initial datum has uniform decay or not and, if there is uniform decay, what are
the upper and lower bounds for these rates.
In this note we only present the results and give ideas of the proofs. The details canbe found in [6]. The main basis for the proofs are:$L^{2}$

Mathematical Analysis of Viscous Incompressible
This work was presented at the RIMS Workshop
Fluid, held in Kyoto, Japan, November 25-27, 2013.
C.J. Niche acknowledges financial support from PRONEX E-26/110.560/2010-APQI, FAPERJ-CNPq
and Ci\^encia sem Fronteiras PVE $0’11/12$ . M. E. Schonbek was partially supported by NSF Grant
DMS-0900909.
–

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(1) “Behavior of solutions for large time is determined by low frequencies of thesolutions” ;(2) Use a time depending filter to study the low frequencies, this is the Fourier
Splitting method [9], [10].

1.1. Background. For the heat equation in
it is very easy to see that the decay
depends on the behavior of the data near the origin in frequency space. For completenesswe describe what happens for such solutions. Let $u=u(x, t)$ be a solution to the heat$\mathbb{R}^{n}$equation$u_{t}-\triangle u=0, u_{0}(x)=u(x, 0)$.

Then$u(x, t)=G_{t}*u_{0}(x)= \frac{1}{(4\pi t)^{\frac{n}{2}}}e^{-\frac{|x|^{2}}{4t}}*u_{0}(x)$.

We study now the diﬀerent possible decay rates.
1,1.1.

Exponential decay. Let$u_{0}\in L^{2}(\mathbb{R}^{n})$and$\hat{u_{0}}(\xi)=0$, when$|\xi|<\delta$. Then$\Vert\hat{u}(t)\Vert_{L^{2}}^{2}=\int_{|\xi|>\delta}e^{-8\pi|\xi|^{2}t}|\hat{u_{0}}(\xi)|^{2}d\xi\leq Ce^{-8\pi\delta^{2}t}.$1.1.2. Slow decay. Let
. Let
then
However, the norm of the gradient scales as, so when getssmaller, the norm of the gradient gets smaller too, so the right hand side of (E) producesslow decay. Namely, for any fixed $t>0$ , decay for solutions with data
will not be
uniform, as$\mathcal{B}=\{v:\Vert v\Vert_{L^{2}}=1\}$$u_{0}^{\lambda}(x)=\lambda^{\frac{n}{2}}e^{-\pi\frac{|\lambda x|^{2}}{2}}$$u_{0}^{\lambda}(x)\in \mathcal{B}.$)$\Vert\nabla u_{0}^{\lambda}\Vert_{L^{2}}=\pi\lambda\Vert\nabla u_{0}\Vert_{L^{2}}$$\lambda$$u_{0}^{\lambda}\in \mathcal{B}$$\frac{\Vert\hat{u^{\lambda}}(t)\Vert_{L^{2}}^{2}}{\Vert\hat{u_{0}^{\lambda}}||_{L^{2}}^{2}}=\frac{1}{1+4\lambda^{2}t}arrow^{\lambda\vec {}0}1.$

So, there exist solutions to the heat equation with data inslowly.
Proposition 1.1. Given $T,$
, there exists
solution to the heat equation with that initial datum.
$r,$$\epsilon>0$$u_{0}$

with$L^{2}(\mathbb{R}^{n})$decaying arbitrarily$\Vert u_{0}\Vert_{L^{2}}=r$

so that for the$\frac{\Vert u(T)\Vert_{L^{2}}}{\Vert u_{0}||_{L^{2}}}\geq 1-\epsilon.$

1.1.3. Algebraic decay. This is easily seen whensee [1].

$|\hat{u}_{0}(\xi)|\approx C|\xi|^{k}$and when $|\hat{u}_{0}(\xi)|\geq C>0,$1.2. Ideas for characterizing decay decay. To characterize theand/or Sobolev
decay of solutions to dissipative
we will:(1) characterize the initial data;(2) understand behavior of solution to the underlying linear equation;(3) study influence of the non linear part;(4) study of diﬀerence of linear and nonlinear solutions;(5) use a reverse triangle inequality to obtain lower bounds of rates of decay.
$L^{2}$$equations_{\rangle}$

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2. CHARACTERIZATION OF THE INITIAL DATUM$s$In this section we introduce the definitions of the decay indicator, decay character,-decay indicator and -decay character.
$s$2.1. Definitions.
Definition 2.1. ([1], [6]) Let
is defined by$u_{0}\in L^{2}(\mathbb{R}^{n})$, $r\in$$(- \frac{n}{2}, \infty)$. The decay indicator$P_{r}(u_{0})$of$u_{0}$$P_{r}(u_{0})= \lim_{\rhoarrow 0}\rho^{-2r-n}\int_{B(\rho)}|\hat{u_{0}}(\xi)|^{2}d\xi,$

where $B(\rho)=\{\xi : |\xi|\leq\rho\}.$Remark 2.1. The decay indicator compareswith $f(\xi)=|\xi|^{r}$ at $\xi=0.$$|\hat{u_{0}}(\xi)|$$\square$. The decay character of is $r^{*}=r^{*}(u_{0})$ , theDefinition 2.2. ([1], [6]) Let
such that $0<P_{r}(u_{0})<\infty$ , if this number exists. If it does not exist
unique $r\in$then$u_{0}\in L^{2}(\mathbb{R}^{n})$$u_{0}$$(- \frac{n}{2}, \infty)$$r^{*}(u_{0})=\{\begin{array}{ll}- \frac{n}{2}, if P_{r}(u_{0})=\infty, for all r\in (- \frac{n}{2}, \infty)\infty, ifP_{r}(u_{0})=0, for all r\in (-\frac{n}{2}, \infty) .\end{array}$Definition 2.3. ([6]) Let
is defined asof$u_{0}\in L^{2}(\mathbb{R}^{n})$$P_{r}^{s}(u_{0})$, $s>0,$ $r\in$$(- \frac{n}{2}+s, \infty)$. The -decay indicator$s$$\Lambda^{s}u_{0}$$P_{r}^{s}(u_{0})= \lim_{\rhoarrow 0}\rho^{-2r-n}\int_{B(\rho)}|\xi|^{2s}|\hat{u_{0}}(\xi)|^{2}d\xi$

where $B(\rho)=\{\xi : |\xi|\leq\rho\}.$Remark 2.2. The -decay indicator compares$s$$|\Lambda^{s}u_{0}(\xi)|$

with $f(\xi)=|\xi|^{r}$ at $\xi=0.$$\square$, the unique $r\in$isDefinition 2.4. ([6]) The -decay character ofsuch that $0<P_{r}^{s}(u_{0})<\infty$ , provided this number exists. If it does not exist
then
$s$

$\Lambda^{s}u_{0}$$r_{s}^{*}=r_{s}^{*}(u_{0})$$(- \frac{n}{2}+s, \infty)$$r_{s}^{*}(u_{0})=\{\begin{array}{ll}\infty, ifP_{r}(u_{0})=0, for all q\in (- \frac{n}{2}+s, \infty)-\frac{n}{2}+s, if P_{r}(u_{0})=\infty, for all q\in(-\frac{n}{2}+s, \infty) .\end{array}$Remark 2.3. If, $1\leq p\leq 2$ , then $r^{*}(u_{0})=-n(1- \frac{1}{p})$ . So, if, withbut is not in any, then $r^{*}(u_{0})=0$ . If

$u_{0}\in L^{p}(\mathbb{R}^{n})\cap L^{2}(\mathbb{R}^{n})$$u_{0}\in L^{1}(\mathbb{R}^{n})\cap L^{2}(\mathbb{R}^{n})$$1\leq p<2$ , then $r^{*}(u_{0})=- \frac{n}{2}.$$u_{0}\in L^{2}(\mathbb{R}^{n})$$L^{p}(\mathbb{R}^{n})$$\square$33

2.2. Results. Here we state a theorem that shows the relation between the decay character and the -decay character. Heuristically, if
is likenear $\xi=0$ , thenmust be likenear $\xi=0$ . Then the decay characterand the -decay charactershould be related through $r^{*}+s=r_{s}^{*}.$$s$$\hat{u_{0}}(\xi)$$|\xi|^{r+s}$$|\xi|^{\prime r}$$\Lambda^{s}u_{0}(\xi)$$r^{*}(u_{0})$$s$$r_{s}^{*}(u_{0})$Theorem 2.5. (Theorem 2.11, [6]) Let, $s>0.$(1) If$then- \frac{n}{2}+s<r_{s}^{*}(u_{0})<\infty$ and $r_{s}^{*}(u_{0})=s+r^{*}(u_{0})$ .
(2)
if and only if $r^{*}(u_{0})=\infty.$(3)
if and only if $r_{s}^{*}(u_{0})=r^{*}(u_{0}^{\sim})+s=- \frac{n}{2}+s.$$u_{0}\in H^{s}(\mathbb{R}^{n})$$\frac{n}{2}<r^{*}(u_{0})<\infty$$r_{s}^{*}(u_{0})=\infty$$r^{*}(u_{0})=- \frac{n}{2}$3. LINEAR PART: EXAMPLES AND DECAY
3.1. Linear Part. Let :space $X$ for which the symbol$\mathcal{L}$be a pseudodiﬀerential operator on a Hilbert
is such that$X^{n}arrow(L^{2}(\mathbb{R}^{n}))^{n}$$\mathcal{M}(\xi)$of$\mathcal{L}$$\mathcal{M}(\xi)=P^{-1}(\xi)D(\xi)P(\xi) , \xi-a.e.$

where $P(\xi)\in O(n)$ and, for $c_{i}>c>0$ and $0<\alpha\leq 1$ . The Laplacianand the fractional Laplacian are examples of such operators.
Given the linear equation$D(\xi)=-c_{i}|\xi|^{2\alpha}\delta_{ij}$$\partial_{t}v=\mathcal{L}v$

multiplying by , integrating in space and using properties of$v$$\mathcal{L}$

we obtain$\frac{d}{dt}\Vert v(t)\Vert_{L^{2}(\mathbb{R}^{n})}^{2}\leq-C\int_{\mathbb{R}^{n}}|\xi|^{2\alpha}|\hat{v}(\xi, t)|^{2}d\xi,$which is inequality (E) .
Example 3.1. Temam [11] introduced the following compressible approximation to NavierStokes equations(3.1)$u_{t}= \mathcal{L}u=\Delta u+\frac{1}{\epsilon}\nabla divu, \epsilon>0$

where the relation-divu eliminates the nonlocal relation between the pressure
and the velocity. The symbol for this operator is, with$\epsilon p=$$( \mathcal{M}(\xi))_{ij}=-|\xi|^{2}\delta_{ij}-\frac{1}{\epsilon}\xi_{i}\xi_{j}$$D( \xi)=diag(-|\xi|^{2}, -|\xi|^{2}, -(1+\frac{1}{\epsilon})|\xi|^{2})$and$P( \xi)=(\frac{}{}\frac{-\xi_{2}}{\sqrt{\xi_{1}^{2}+\xi_{2}^{2}}\sqrt{\xi^{2}+\xi_{2}^{2}\not\in},01} \frac{--\sqrt{1-\xi_{3}^{2}}^{L3}\sqrt{1-\xi_{3}^{2}}^{\Delta}-\xi_{23}1-\xi^{2}}{\sqrt{1-\xi_{3}^{2}}} \xi_{3}\xi_{2}\xi_{1})$Then, the kernel is given by(3.2)

see Rusin [8].

$(e^{t\mathcal{M}(\xi)})_{ij}=e^{-t|\xi|^{2}} \delta_{ij}-\frac{\xi_{i}\xi_{j}}{|\xi|^{2}}(e^{-t|\xi|^{2}}-e^{-(1+\frac{1}{\epsilon})t|\xi|^{2})},$

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3.2. Decay of linear part. In this subsection we give the main decay theorems for the
linear equations and give the sketch of some of the proofs.
have decay character $r^{*}(v_{0})=r^{*}$
Theorem 3.2. (Theorem 2.10, [6]) Let
Let $v(t)$ be a solution to the linear equation with initial datum . Then:$C_{2}>0$ such that, there exist constants(1) if$v_{0}\in L^{2}(\mathbb{R}^{n})$$v_{0}$$C_{1},$$\frac{n}{2}<r^{*}<\infty$$C_{1}(1+t)^{-\frac{1}{\alpha}(_{\tau+r}^{n})}\leq\Vert v(t)\Vert_{L^{2}}^{2}\leq C_{2}(1+t)^{-\frac{1}{\alpha}(\frac{n}{2}+r^{*})}$(2) if$r^{*}=- \frac{n}{2}$;, there exists $C=C(\epsilon)>0$ such that$\Vert v(t)\Vert_{L^{2}}^{2}\geq C(1+t)^{-\epsilon}, \forall\epsilon>0,$

is slower than any uniform algebraic rate;(3) if $r^{*}=\infty$ , there exists $C>0$ such that$i.e$

. the decay of$\Vert v(t)\Vert_{L^{2}}^{2}$$\Vert v(t)\Vert_{L^{2}}^{2}\leq C(1+t)^{-m}, \forall m>0,$$i.e$. the decay of$\Vert v(t)\Vert_{L^{2}}$is faster than any algebraic rate.

Proof We only sketch the proof of (1). We first prove lower bounds for decay. Using
the properties of the symbol of , we obtain$\mathcal{L}$$|e^{\mathcal{M}(\xi)t}\hat{v_{0}}(\xi)|\geq Ce^{-ct\rho^{2\alpha}(t)}|\hat{v_{0}}(\xi)|.$

Then$\Vert v(t)\Vert_{L^{2}}\geq C\rho^{2r+n}e^{-ct\rho^{2\alpha}(t)}\rho^{-2r-n}\int_{B(\rho(t))}|\hat{v_{0}}(\xi)|^{2}d\xi$$\geq C\rho^{2r+n}e^{-t\rho^{2\alpha}(t)}.$Choosing$\rho(t)=\rho_{0}(1+t)^{-\frac{1}{2\alpha}}$

we have$\Vert v(t)\Vert_{L^{2}}^{2}\geq C(1+t)^{-\frac{1}{\alpha}(\frac{n}{2}+r^{*})}.$For the upper bounds, from$\frac{d}{dt}\Vert v(t)\Vert_{L^{2}}^{2}\leq-C\rho^{2\alpha}(t)\int_{B^{c}(\rho(t))}|\hat{v}(\xi)|^{2}d\xi,$using the Fourier Splitting method we obtain$\frac{d}{dt}\Vert v(t)\Vert_{L^{2}}^{2}+\rho^{2\alpha}(t)\Vert v(t)\Vert_{L^{2}}^{2}\leq C\rho^{2\alpha}(t)\int_{B(t)}|\hat{v}(\xi)|^{2}d\xi.$

As $P_{r}(u_{0})<\infty$ , there are$\rho_{0}>0,$$C>0$ such that for $0<\rho<\rho_{0}$$\rho^{-2r-n}\int_{B(\rho(t))}|\hat{v_{0}}(\xi)|^{2}d\xi\leq C.$

Then$\frac{d}{dt}\Vert v(t)\Vert_{L^{2}}^{2}\rho^{2\alpha}(t)\leq C\rho^{2\alpha+2r+n}(t)$.

35and choosing$\rho(t)=m^{\frac{1}{2\alpha}}(1+t)^{-\frac{1}{2\alpha}}$, with $m>r+ \frac{n}{2}$ and using the integrating factor$h(t)=(1+t)^{m}$ , we obtain$\frac{d}{dt}((1+t)^{m}\Vert v(t)\Vert_{L^{2}}^{2})\leq C(1+t)^{m-1-\frac{r}{\alpha}-\frac{n}{2\alpha}}.

\square$For the decay of derivatives in the case of solutions to linear equations we have thefollowing result.
Theorem 3.3. (Theorem 2.12, [6]) Let, with $s>0$ , have decay character. Then the solution of the linear equation with datum satisfies:(1) if$C_{2}>0$ such that, there exist constants$v_{0}\in H^{s}(\mathbb{R}^{n})$$r_{s}^{*}=r_{S}^{*}(v_{0})$$v_{0}$$\frac{n}{2}\leq r^{*}<\infty$$C_{1},$$C_{1}(1+t)^{-\frac{1}{\alpha}(\frac{n}{2}+r^{*}+s)}\leq\Vert v(t)\Vert_{\dot{H}^{s}}^{2}\leq C_{2}(1+t)^{-\frac{1}{\alpha}(\frac{n}{2}+r^{*}+s)}$;(2) if $r^{*}=\infty$ , then$\Vert v(t)\Vert_{\dot{H}^{s}}^{2}\leq C(1+t)^{-r}, \forall r>0,$$i.e$. the decay of$\Vert v(t)\Vert_{\dot{H}^{s}}$is faster than any algebraic rate.

4. APPLICATIONS
4.1. Quasi-Geostrophic equations. In this section we study the upper and lower decay
rates for solutions to the Dissipative Quasi-Geostrophic (DQGE) equation. The DQGE
is given by$\theta_{t}+u\cdot\nabla\theta+(-\Delta)^{\alpha}\theta=0, 0<\alpha\leq 1$

where$\theta$

is the potential temperature of a fluid in$\mathbb{R}^{2}$, and$u=R^{\perp}\theta=(-R_{2}\theta, R_{1}\theta)$

is its velocity, whereis the Riesz transform in . This equation models importantgeophysical phenomena ([5], [7]) and when
it provides a good model for $3D$ NavierStokes equations ([2], [4]).
$R_{i}$$x_{i}$$\alpha=\frac{1}{2}$4.1.1. Results for DQGE. The followipg Theorem summarizes all our results.
Theorem 4.1. ([6]) Let, with decay character $r^{*}=r^{*}(u_{0})$ .
(1) If, then there exists constants$C_{2}>0$ so that$\theta_{0}\in L^{2}(\mathbb{R}^{2})$$r^{*}\leq 1-\alpha$$C_{1},$$C_{1}(1+t)^{-\frac{1}{\alpha}(1+r^{*})}\leq\Vert\theta(t)\Vert_{L^{2}}^{2}\leq C_{2}(1+t)^{-\frac{1}{\alpha}(1+r^{*})}$(2) If $r^{*}\geq 1-\alpha,$ $r^{*} \leq\min\{1, 2(1-\alpha)\}$ then there exist$C_{1},$$C_{2}>0$$C_{1}(1+t)^{-\frac{1}{\alpha}(1+r^{*})}\leq\Vert\theta(t)\Vert_{L^{2}}^{2}\leq C_{2}(1+t)^{-(2-\alpha)}\tilde{\alpha};1$(3) If $r^{*}>1$ and $r^{*}\geq 2(1-\alpha)$ we have that$\Vert\theta(t)\Vert_{L^{2}}^{2}\leq C_{2}(1+t)^{-\frac{1}{\alpha}(2-\alpha)}.$We now give partial statements and sketches of proofs.

;

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, let
Theorem 4.2. (Theorem 3.1, [6]) Let$0<\alpha\leq 1$ . Let
be a weak solution to $QGE$ with data . Then:(1) If $r^{*}\leq 1-a$ , then$r^{*}=r^{*}(\theta_{0})$$\theta_{0}\in L^{2}(\mathbb{R}^{2})$$,$$-1<r^{*}<\infty$ ,and$\theta_{0}$$\theta$$\Vert\theta(t)\Vert_{L^{2}}^{2}\leq C(t+1)^{-\frac{1}{\alpha}(1+r)}$;(2) if $r^{*}\geq 1-\alpha$ , then$\Vert\theta(t)\Vert_{L^{2}}^{2}\leq C(t+1)^{-\frac{1}{\alpha}(2-\alpha)}.$Proof We give formal estimates, which have to be proven rigurously by taking approximations and passing to the limit. Let$B(t)= \{\xi\in \mathbb{R}^{2}:|\xi|^{2\alpha}\leq\frac{f’(t)}{2f(t)}\}.$

Through the Fourier Splitting method we obtain$\frac{d}{dt}(f(t)\Vert\theta(t)\Vert_{L^{2}}^{2})\leq f’(t)\int_{B(t)}|\hat{u}(\xi, t)|^{2}d\xi$(4.3)$\leq Cf’(t)(\Vert\Theta(t)\Vert_{L^{2}}^{2}+\int_{B(t)}(\int_{0}^{t}e^{-(t-s)|\xi|^{2\alpha}}|\xi||\hat{u\theta}(\xi, s)|^{2}ds)^{2}d\xi)$,is the solution to the linear part. First, we obtain a preliminary decay by
where$f(t)=[\ln(e+t)]^{1+\frac{1}{\alpha}},$ $0<\alpha<1$ or $f(t)=[\ln(e+t)]^{3}$ , for $\alpha=1$ . In this casechoosing$\Theta=$$\Vert\theta(t)\Vert_{L^{2}}^{2}\leq\Vert\Theta(t)\Vert_{L^{2}}^{2}+C[\ln(e+t)]^{-(1+\frac{1}{\alpha})}\leq C[\ln(e+t)]^{-(1+\frac{1}{\alpha})}.$

Now we bootstrap with the new choice $f=(t+1)^{\beta},$
we obtain
decay in (4.3) and dividing by$\beta\gg 1$. Plugging in the preliminary$(t+1)^{\beta-\frac{2}{\alpha}+1}$$(t+1)^{\frac{2}{\alpha}-1}\Vert\theta(t)\Vert_{L^{2}}^{2} \leq \Vert\theta_{0}\Vert_{L^{2}}^{2}(t+1)^{-(\beta-\frac{2}{\alpha}+1)}+C(t+1)^{\frac{1}{\alpha}-\frac{r^{*}}{\alpha}-1}$$+ C \int_{0}^{t}\frac{(s+1)^{1-\frac{2}{\alpha}}}{\ln(e+s)]^{1+\frac{1}{\alpha}}}(s+1)^{\frac{2}{\alpha}-1}\Vert\theta(s)\Vert_{L^{2}}^{2}ds.$

Define$\psi(t)=(1+t)^{\frac{2}{\alpha}-1}\Vert\theta(t)\Vert_{L^{2}}^{2}, a(t)=C(t+1)^{-\beta}+(1+t)^{\frac{1}{\alpha}-\frac{f}{\alpha}-1},$$b(t)=C[\ln(e+t)]^{-(1+\frac{1}{\alpha})}(s+1)^{1-\frac{2}{\alpha}}.$and then use Gronwall’s inequality to obtain final estimates.
For the derivatives the decay result is the following.
Theorem 4.3. (Theorem 3.5, [6]) Let
the solutions to $QGE$ satisfy$\frac{1}{2}<\alpha\leq 1,$$\square .$$\alpha\leq s$$r^{*}=r^{*}(\theta_{0})$(1) if $r^{*}\leq 1-\alpha$ , then$\Vert\theta(t)\Vert_{\dot{H}^{s}}^{2}\leq C(t+1)^{-\frac{1}{\alpha}(s+1+r^{*})}$(2) if $r^{*}\geq 1-\alpha$ , then$\Vert\theta(t)\Vert_{\dot{H}^{s}}^{2}\leq C(t+1)^{-\frac{1}{\alpha}(s+2-\alpha)}.$;and$\theta_{0}\in H^{s}(\mathbb{R}^{2})$

. For

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We now state the Theorem that deals with the decay of the diﬀerence between thelinear and the nonlinear parts.

Theorem 4.4. (Theorem 3.2, [6]) Let $0<\alpha\leq 1,$$\theta_{0}\in L^{2}(\mathbb{R}^{2})$. Then(1) $if-1<r^{*}\leq\alpha-l$ then$|\}\theta(t)-\Theta(t)\Vert_{L^{2}}^{2}\leq C(1+t)^{-\frac{1}{\alpha}(2-\alpha+r^{*})}$;(2) if $\alpha-1<r^{*}\leq 1-\alpha$ then$\Vert\theta(t)-\Theta(t)\Vert_{L^{2}}^{2}\leq C(1+t)^{-\frac{1}{\alpha}\min\{2,2-\alpha+r^{*}\}}$;(3) if $r^{*}\geq 1-\alpha$ , then$\Vert\theta(t)-\Theta(t)\Vert_{L^{2}}^{2}\leq C(1+t)^{-\frac{1}{\alpha}\min\{3-2\alpha,2\}}.$

Proof The main term to estimate is$| \int_{\mathbb{R}^{2}}\Theta(u\cdot\nabla\theta)dx|\leq\Vert\nabla\Theta(t)\Vert_{\infty}\Vert\theta(t)\Vert_{2}^{2}\leq C(1+t)^{-\gamma}=h(t)$.

For the proof of this theorem we use Fourier splitting and follow the ideas in [3].
The bounds for the diﬀerence between the linear solution and the nonlinear one, combined with the bounds of the linear solution yields the lower bounds of decay.
$\square$

Theorem 4.5. (Theorem 3.3, [6]) Let $0<\alpha\leq 1,$$0< \alpha\leq\frac{1}{2}$$and-1<r^{*}\leq 1$

or$\frac{1}{2}<\alpha\leq 1$$\theta_{0}\in L^{2}(\mathbb{R}^{2})$,$r^{*}=r^{*}(\theta_{0})$$and-1<r^{*}\leq 2(1-\alpha)$. Then, for

we have that$\Vert\theta(t)\Vert_{L^{2}}^{2}\geq C(1+t)^{-\frac{1}{\alpha}(1+r^{*})}.$

Proof Follows from the reverse triangle inequality$\Vert\theta(t)\Vert_{L^{2}}^{2}\geq\Vert\Theta(t)\Vert_{L^{2}}^{2}-\Vert\theta(t)-\Theta(t)||_{L^{2}}^{2},$provided that the linear part has slower decay than the diﬀerence between the solutions
and the linear part.
$\square .$4.2. Approximation for compressible Navier-Stokes. In the Navier-Stokes equations, the pressure is a nonlocal function of the velocity. This poses important problems
when using numerical methods to study solutions to this system. Temam [11] introduced
a compressible approximation to Navier-Stokes by relating the pressure and the velocity
through. In order to have an energy inequality, he stabilized the equation byintroducing a term of the form
. Then, the system obtained is$\nabla\cdot u=-\epsilon p$$\frac{1}{2}(divu)u$$u_{t}^{\epsilon}+(u^{\epsilon} \cdot\nabla)u^{\epsilon} + \frac{1}{2}(divu^{\epsilon})u^{\epsilon}=\triangle u^{\epsilon}+\frac{1}{\epsilon}\nabla divu^{\epsilon},$$u_{0}^{\epsilon}(x) = u^{\epsilon}(x, 0)$

The linear part of this system, i.e.

.

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$u_{t}= \mathcal{L}u = \Delta u+\frac{1}{\epsilon}\nabla.

divu=0,$$( \mathcal{M}_{\epsilon}(\xi, t))_{kl}=e^{-t|\xi|^{2}}\delta_{kl} - \frac{\xi_{k}\xi_{l}}{|\xi|^{2}}(e^{-t|\xi|^{2}}-e^{-(1+\frac{1}{\epsilon})t|\xi|^{2}})$,fits exactly in our setting, see Example 3.1. As the nonlinear part$(u \cdot\nabla)u+\frac{1}{2}$(divu) $u= \nabla(u\otimes u)-\frac{1}{2}$ (divu)$u.$

has a structure similar to Navier-Stokes and the compressible part can be easily handledsince$\int_{\mathbb{R}^{3}}u(u\cdot\nabla)udx=-\frac{1}{2}\int_{\mathbb{R}^{3}}|u|^{2}divudx,$

we have the energy inequality of the form (E) .
in
Remark 4.4. Rusin [8] proved existence of weak solutions to these system withconvergezero,
tosuitablegoes
thesolutionsto. He also proved that whensolutions of the Navier-Stokes system.
$u_{0}^{\epsilon}$$L^{2}(\mathbb{R}^{3})$$\epsilon$$\square$4.2.1. Results for the compressible approximation to Navier-Stokes. Here we just list theresults and for details refer the reader to [6]. The methods and techniques for the proofs
are similar to those for the DQGE.
Theorem 4.6. ([6]) Let$C_{2}>0$ such that$u_{0}^{\epsilon}\in L^{2}(\mathbb{R}^{3})$, $r^{*}=r^{*}(u_{0})$ . Then $for- \frac{3}{2}<r^{*}\leq 1$ , there exist$C_{1},$$C_{1}(1+t)^{-(\frac{3}{2}+r)}\leq\Vert u^{\epsilon}(t)\Vert_{L^{2}}^{2}\leq C_{2}(1+t)^{-(\S+r^{e})}.$

If $r^{*}>1$ , then$\Vert u^{\epsilon}(t)\Vert_{L^{2}}^{2}\leq C(1+t)^{-\frac{6}{2}}.$

Theorem 4.7. (Theorem 3.14, [6]) Let$1\leq s\leq r$ we have that$u_{O}\in H^{r}(\mathbb{R}^{3})$, $r\geq 1,$ $r^{*}=r^{*}(u_{0})$ . Then, for$\Vert u(t)\Vert_{\dot{H}^{\epsilon}}\leq C(1+t)^{-\frac{1}{2}(s+\min\{\frac{3}{2},r+\frac{3}{2}\})}.$

Theorem 4.8. (Theorem 3.10, [6]) Let $\epsilon>0,$. Then$u_{0}^{\epsilon}\in L^{2}(\mathbb{R}^{3})$, and $r^{*}=r^{*}(u_{0})$ with$- \frac{3}{2}<r^{*}<\infty$$\Vert u^{\epsilon}(t)-\overline{u}(t)\Vert_{L^{2}}^{2}\leq C(1+t)^{-\min\{\frac{7}{4},\frac{7}{4}+r^{*}\}}$

Theorem 4.9. (The\‘orem 3.11, [6]) Let
we have that$u_{0}^{\epsilon}\in L^{2}(\mathbb{R}^{3})$, $r^{*}=r^{*}(u_{0})$ . Then for-$\Vert u^{\epsilon}(t)\Vert_{L^{2}}^{2}\geq C(1+t)^{-(\frac{3}{2}+r^{*})}.$$\frac{3}{2}<r^{*}\leq 1$

39Remark 4.5. The estimates for this compressible approximation are the same as thoseobtained by Bjorland and M.E. Schonbek [1] for the Navier-Stokes equations. Hence, thestabilizing nonlinear damping term
provides enough dissipation to have anenergy inequality, but does not alter the range of values offor which the linear part$\frac{1}{2}(divu^{\epsilon})u^{\epsilon}$$r^{*}$

has slower decay.

$\square$

5. FINAL COMMENTS(1) The decay character classifies thedata for dissipative equations, at least when
the linear part has slower decays. The linear part has to be studied first, then$L^{2}$

the whole nonlinear system.
(2) We are able to obtain information on both upper and lower decay rates, sometimes
sharply characterizing the decay in terms of the initial data.

REFERENCES[1] Clayton Bjorland and Maria E. Schonbek. Poincar\’e’s inequality and diﬀusive evolution equations.

Adv. Diﬀerential Equations, $14(3-4):241-260$ , 2009.
[2] Luis A. Caﬀarelli and Alexis Vasseur. Drift diﬀusion equations with fractional diﬀusion and thequasi-geostrophic equation. Ann. of Math. (2), $171(3):1903-1930$ , 2010.
[3] Peter Constantin and Jiahong Wu. Behavior of solutions of 2D quasi-geostrophic equations. SIAM
J. Math. Anal., $30(5):937-948$ , 1999.
[4] A. Kiselev, F. Nazarov, and A. Volberg. Global well-posedness for the critica12D dissipative quasigeostrophic equation. Invent. Math., $167(3):445-453$ , 2007.
[5] Andrew J. Majda and Esteban G. Tabak. A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow. Phys. D, $98(2-4):515-522$ , 1996. Nonlinear phenomena
in ocean dynamics (Los Alamos, NM, 1995).
[6] C\’esar J. Niche and Mar\’ia E. Schonbek. Decay characterization of solutions to dissipative equations.
ArXiv e-prints:math.AP/1405.7565, May 2014.
[7] Joseph Pedlosky. Geophysical Fluid Dynamics. Springer, New York, 1987.
[S] Walter Rusin. Incompressible 3d Navier–Stokes equations as a limit of a nonlinear parabolic system.
Journal of Mathematical Fluid Mechanics, $14(2):383-405$ , 2012.
[9] Mar\’ia E. Schonbek.
decay for weak solutions of the Navier-Stokes equations. Arch. Rational
Mech. Anal., $88(3):209-222$ , 1985.
[10] Mar\’ia E. Schonbek. Large time behaviour of solutions to the Navier-Stokes equations. Comm. Partial
Diﬀerential Equations, 11 (7):733-763, 1986.
[11] Roger Temam. Une m\’ethode d’approximation de la solution des \’equation de Navier-Stokes. Bull.
Soc. Math. FVance, 96:115-152, 1968.
$L^{2}$(C.J. Niche) DEPARTAMENTO DE MATEM\’ATICA APLICADA, INSTITUTO DE MATEM\’ATICA, UNIVERSIDADE
FEDERAL DO RIO DE JANEIRO, CEP 21941-909, RIO DE JANEIRO-RJ, BRASIL$E$-mail address:
cniche@im.ufrj. br(M.E. Schonbek) DEPARTMENT OF MATHEMATICS, UC SANTA CRUZ, SANTA CRUZ, CA 95064, USA$E$-mail address: schonbekQucsc. edu