数理解析研究所講究録
第 1782 巻 2012 年 44-50

44

Fundamental solutions of diffusion equations
related to certain Dirichlet forms
and the quasi-geostrophic equation
Yasunori Maekawa
Department of Mathematics, Graduate School of Science, Kobe University
1-1 Nada-ku, Rokkodai, Kobe 657-8501, Japanyasunori@math.kobe-u.ac.jp

Hideyuki Miura
Department of Mathematics, Graduate School of Science, Osaka University
1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japanmiura@math.sci.osaka-u.ac.jp

1

Introduction

We consider the non-local diffusion equations in the presence of a drift term$\partial_{t}\theta+A_{K}(t)\theta+v\cdot\nabla\theta=0$,$t>0$ ,

$x\in \mathbb{R}^{d}$

(1.1)

,

where $d\geq 2$ and $A_{K}(t)$ is a linear operator formally defined by$(A_{K}(t)f)(x)=P.V \int_{R^{d}}(f(x)-f(y))K(t, x, y)dy$.

(1.2)

Here $K(t, x, y)$ is a positive function satisfying $K(t, x, y)=K(t, y, x)$ and $v(t, x)=$$(v_{1}(t, x), \cdots, v_{d}(t, x))$ is a vector field insatisfying the divergence free condition,$A_{K}(t)$$\nabla\cdot v(t)=0$ .
will be supposed to possess a diffusion effect likeIn particular,
with $\alpha\in(0,2)$ thefor some $\alpha\in(0,2)$ . Note that in the case
$K$
$K(t,
x,
y)=C_{d,\alpha}|x-y|^{-d-\alpha}$. The
positive
kernelis given by
constant
for some
aim of this note is to show the existence and the continuity of fundamental solutions for(1.1) under less regularity assumptions on $K$ and .
$\mathbb{R}^{d}$$A_{K}(t)=(-\triangle)^{\alpha/2}$$(-\triangle)^{\alpha/2}$$C_{d,\alpha}$

$v$

45
When there is no drift term $(i.e., v=0)$ this problem appears in the theory of Dirichletforms of jump type. For the diffusion operator $A_{K}(t)$ defined by (1.2) the associatedDirichlet form is$\mathcal{E}_{K}^{(t)}(f, g)=\frac{1}{2}ll_{]R^{2d}}[f][g](x, y)K(t, x, y)dxdy$$[f](x, y)=f(x)-f(y)$ ,

,

(1.3)

and it has been investigated mainly from the probabilistic approach [6, 16, 17, 3, 1, 13, 2].
On the other hand, in recent years the case with a drift term has also attracted muchattention, especially in the field of fluid mechanics, mathematical finance, biology, and soon. For example, many works have been done for the two-dimensional dissipative quasigeostrophic equations (QG), where $A_{K}(t)=(-\triangle)^{\alpha/2}$ and the drift term is a nonlinear
term such that is given in terms of via the Riesz transform; [7]. For such nonlinear
problems it is crucial to obtain detailed informations of solutions under less regularity$v$$\theta$conditions on .
In [4, 12] fundamental solutions were constructed whenwith$(1,2)$ andbelongs to a suitable Kato class without assuming the divergence free condition. In this case the diffusion term is the leading term and they showed two-sided heat
kernel estimates by using perturbation arguments. However, despite of the increasing
interest, there seems to be still few works on fundamental solutions for $\alpha\in(0,1]$ . In suchcases the drift term formally becomes the leading term and is no longer regarded as asimple perturbation of $A_{K}(t)$ , which causes difficulties in the study of (1.1). For example,
so far little seem to be known about the uniqueness of weak solutions for such cases andthis makes even the semigroup property of fundamental solutions nontrivial.
To state our main results we give the precise assumptions on the kernel $K$ and thevelocity . We assume that there are $\alpha\in(0,2)$ and $C_{0}>0$ such that$v$

$A_{K}(t)=(-\triangle)^{\alpha/2}$$\alpha\in$

$v$

$v$

$K(t, x, y)=K(t, y, x)$$ess.\sup_{t>0,x\in R^{d}}\int_{|x-y|\leq M}|x-y|^{2}K(t, x, y)dy$$ess.\inf_{t>0,x,y\in R^{d}}|x-y|^{d+\alpha}K(t, x, y)$for $a.e$ .

$(t, x, y)\in(O, \infty)\cross \mathbb{R}^{d}\cross \mathbb{R}^{d},$$\leq C_{0}M^{2-\alpha}$$\geq C_{0}^{-1}$for each$M\in(O, \infty)$$(1.4)$, (1.5)(1.6).

Following the conventions in (QG), we will call the case $\alpha\in(1,2)$ subcritical, the casecritical, and the case $\alpha\in(0,1)$ supercritical. We note that if $K$ satisfies$\alpha=1$$C^{-1}|x-y|^{-d-\alpha}\leq K(t, x, y)\leq C|x-y|^{-d-\alpha}$

,

(1.7)

then (1.5) and (1.6) are satisfied. The condition (1.6) guarantees the diffusion effect like
. Taking this in mind, we assume that belongs to a class of functions which isinvariant under the scaling$(-\triangle)^{\alpha/2}$$v$$v(t, x) r^{1-1/\alpha}v(rt, r^{1/\alpha}x)$This scaling is natural in the following sense: ifand $v=v(t, x)$ then the rescaled function$\theta(t, x)$$(-\triangle)^{\alpha/2}$,$r>0$ .

(1.8)

is a solution to (1.1) with $A_{K}(t)=$satisfies (1.1) with $A_{K}(t)=$$\theta(rt, r^{1/\alpha}x)$

46
and $v=r^{1-1/\alpha}v(rt, r^{1/\alpha}x)$ , instead of $v(t, x)$ . Heuristically it is essential thatbelongs to a function space which is invariant with respect to (1.8), in order to ensure; for example, see [19, 22, 20] and references there in fora smoothing effect byequationsand [5, 14, 21] for the fractional diffusion equations.
second order parabolic
To describe the regularity assumption on let us introduce the Campanato spaces;see [9].
$(-\triangle)^{\alpha/2}$$v$$(-\triangle)^{\alpha/2}$$v$$\mathcal{L}^{p,\lambda}(\mathbb{R}^{d})=\{f\in L_{loc}^{q}(\mathbb{R}^{d})|\Vert f\Vert_{\mathcal{L}^{p,\lambda}}=\sup_{B}(R^{-\lambda}\int_{B}|f(x)-f_{B}f|^{p}dx)^{\frac{1}{p}}<\infty\}$

.

(1.9)

Here the supremum is taken over all balls $B=B_{R}(y)$ (the ball with radius $R>0$ centered
for $B_{R}(0)$, and $|B|$ is the volume of the ball $B$ . We will sometimes writeatis defined byfor simplicity of notations. The value$B_{R}$$y\in \mathbb{R}^{d})$$\#_{B}f$$f_{B}f= \frac{1}{|B|}\int_{B}f(x)dx$(1.10).

Then we haveif

$L_{w}^{d-}p_{\frac{d}{\lambda}}(\mathbb{R}^{d})$$arrow$$\mathcal{L}^{p,\lambda}(\mathbb{R}^{d})$$=$$\mathcal{L}^{p,\lambda}(\mathbb{R}^{d})$$=$$\mathcal{L}^{p,\lambda}(\mathbb{R}^{d})$$BMO(\mathbb{R}^{d})$$L^{p}$$L_{w}^{p}(\mathbb{R}^{d})$ifif$\dot{c}\frac{\lambda-d}{p}(\mathbb{R}^{d})$space andHereis the weak
space of the order .
Next we introduce the Morrey type spaces of$0<\lambda<d$$\dot{C}^{\beta}(\mathbb{R}^{d}),$$\lambda=d$,$d<\lambda\leq d+p$$\beta\in(0,1]$(1.11)(1.12),(1.13).

, is the homogeneous H\"older$\beta$$\mathcal{L}^{p,\lambda}$-valued functions.

$L^{p,\lambda_{1}}(0, \infty;\mathcal{L}^{q,\lambda_{2}}(\mathbb{R}^{d}))=\{f\in L_{loc}^{p}(0, \infty;\mathcal{L}^{q,\lambda_{2}}(\mathbb{R}^{d}))|$$((t-s)^{-\lambda_{1}} \int_{s}^{t}$$\Vert f\Vert_{L^{p,\lambda_{1}}(0,\infty;\mathcal{L}^{q,\lambda_{2}}(R^{d}))}=\sup_{t>00}\sup_{<s<t}$

For $1\leq p,$$q\leq\infty$let$L_{loc}^{q}(0, \infty;L_{loc}^{p}(\mathbb{R}^{d}))$$L_{loc}^{q}(0, \infty;L_{loc}^{p}(\mathbb{R}^{d}))=\{f\in L_{loc}^{1}((0$

, oo)

llf

$(\tau)\Vert_{\mathcal{L}^{q,\lambda_{2}}}^{p}d\tau)^{\frac{1}{p}}<\infty\}.(1.14)$

be the class of functions defined by$\cross \mathbb{R}^{d})|\Vert f\Vert_{L^{q}(0,R;Lp(B_{R}))}<\infty$

for all $R>0\}$ .
(1.15)

When $K(t, x, y)$ satisfies (1.5) and (1.6) for some $\alpha\in(0,2)$ the velocity
satisfy the following two conditions:is assumed to$v$(Cl) there are $\lambda\in[2d/\alpha-d, 2d/\alpha+d]$ and $1<q\leq\infty$ such that$v\in L^{1,\frac{1}{2}+\frac{1}{a}-\frac{\lambda}{2d}}(0, \infty, (\mathcal{L}^{\frac{2d}{\alpha},\lambda}(\mathbb{R}^{d}))^{d})\cap L_{loc}^{q}(0, \infty;(L_{loc}^{p_{\lambda}}(\mathbb{R}^{d}))^{d})$where$p_{\lambda}=1$

if $\lambda\in[2d/\alpha-d, d]$ and$p_{\lambda}=\infty$

if $\lambda\in(d, 2d/\alpha+d]$ .

(C2) $\nabla\cdot v(t)=0$ for a.e. $t>0$ in the sense of distributions.

,(1.16)

47
Remark 1.1 The spaceis invariant under the scaling (1.8).
One of the advantages to use the Campanato spaces (1.9) is that for some exponents
they contain functions growing at spatial infinity. In particular, the case $\lambda=2d/\alpha+d$in (Cl) allows to grow at most linearly as. We also note that the condition
includes the case$L^{1,\frac{1}{2}+\frac{1}{\alpha}-\frac{\lambda}{2d}}(0, \infty;\mathcal{L}^{\frac{2d}{\alpha},\lambda}(\mathbb{R}^{d}))$$(p, \lambda)$$|x|arrow\infty$$v$$v\in L^{1,\frac{1}{2}+\frac{1}{\alpha}-\frac{\lambda}{2d}}(0, \infty;(\mathcal{L}^{\frac{2d}{a},\lambda}(\mathbb{R}^{d}))^{d})$$|t-t_{0}|^{\frac{\lambda}{2d}+\frac{1}{2}-\frac{1}{\alpha}}v(t)\in L^{\infty}(0, \infty;(\mathcal{L}^{\frac{2d}{\alpha},\lambda}(\mathbb{R}^{d}))^{d})$for some$t_{0}\in[0, \infty)$.

(1.17)

Under the divergence free condition (C2) the drift term becomes skew-symmetricwith respect to the usual
inner product, and hence, the adjoint equation for (1.1)
takes the same form as (1.1). This additional structure is essentially used in constructing
fundamental solutions under weak regularity condition (Cl). The divergence free condition sometimes plays important roles also in the second order parabolic equations withsingular drifts; [19, 22, 20].
$L^{2}(\mathbb{R}^{d})$For simplicity of notations we will introduce the seminorm$\Vert v\Vert_{X_{\lambda}}=\Vert v\Vert_{L^{1^{1\lambda}}},z^{+\frac{1}{\alpha}-}\varpi(0,\infty;\mathcal{L}^{\frac{2d}{\alpha},\lambda}(R^{d}))$For $T>0$ and$x\in \mathbb{R}^{d}$$p_{\lambda}$(1.18)

we also set$\Vert v\Vert_{Y_{T,x}^{q,\lambda}}=\Vert v\Vert_{Lq(0,T;L^{p_{\lambda}}(B_{1}(x)))}$

where

.

(1.19),

is as in (Cl).

Let $T>s\geq 0$ . A function
for $t\in[s, T)$ with initial data

is said to be a weak solution to (1.1)
satisfies$\theta\in L^{\infty}(s, T;L^{2}(\mathbb{R}^{d}))$$\theta_{s}$at $t=s$ if$\theta$$\int_{s}^{T}\epsilon_{K}^{(t)}(\theta(t), \theta(t))dt<\infty$,(1.20)

and$l_{s}^{T}(-<\theta(t),$ $\partial_{t}\varphi(t)>+\mathcal{E}_{K}^{(t)}(\theta(t), \varphi(t))-<\theta(t),$ $v(t)\cdot\nabla\varphi(t)>)dt=<\theta_{s},$ $\varphi(s)>$(1.21)

for all, whereis the usual $L^{2}-L^{2}$ pairing in. Then ameasurable function $P_{K,v}(t, x;s, y)$ on $\{(t, s, x, y)|t>s\geq 0, x, y\in \mathbb{R}^{d}\}$ is said to be afundamental solution to (1.1) if for each $T>s\geq 0$ andthe function$\varphi\in C_{0}^{\infty}([s, T)\cross \mathbb{R}^{d})$$<\cdot,$$\mathbb{R}^{d}$$\cdot>$$f\in L^{2}(\mathbb{R}^{d})$$(P_{K,v}f)(t, s, x)$ $:= \int_{R^{d}}P_{K,v}(t, x;s, y)f(y)dy$,

is a weak solution to (1.1) for $t\in[s, T)$ with initial data at $t=s$ .
The main result of this note is the existence of fundamental solutions for (1.1).
$f$(1.22)

48
Theorem 1.2 Suppose that $(1.4)-(1.6)$ and (Cl) $-(C2)$ hold. Then there exists a

fundamental solution $P_{K,v}(t, x;s, y)$ for (1.1) satisfying the following properties.
$\int_{R^{d}}P_{K,v}(t, x;s, y)dx=\int_{R^{d}}P_{K,v}(t, x;s, y)dy=1$$0\leq P_{K,v}(t, x;s, y)\leq C(t-s)^{-\frac{d}{a}}$(1.23),(1.24),,$t>\tau>s\geq 0,$ $(1.25)$$P_{K,v}(t, x;s, y)= \int_{R^{d}}P_{K,v}(t, x;\tau, z)P_{K,v}(\tau, z;s, y)dz$$|P_{K,v}(t, x_{1};s, y_{1})-P_{K,v}(t, x_{2};s, y_{2})| \leq\frac{C’(|x_{1}-x_{2}|^{\beta}+|y_{1}-y_{2}|^{\beta})}{(t-s)^{c}}$(1.26),

and for $T\geq t_{i}>s_{i}\geq 0,$ $i=1,2$ ,(1.27).

$|P_{K,v}(t_{1}, x;s_{1}, y)-P_{K,v}(t_{2}, x;s_{2}, y)| \leq\frac{C_{T,x}|t_{1}-t_{2}|^{\beta’}+C_{T,y}|s_{1}-s_{2}|^{\beta’}}{(\min\{t_{1}-s_{1},t_{2}-s_{2}\})^{c}}$, the positive constants $C’,$, andHere the positive constant depends only on(or) depends only
, the positive constant, anddepend only on
and the positive constants $c’,$, andon $T,$.
depend only on
$C$

$d,$

$d,$

$\alpha,$$C_{0},$$\alpha,$$\lambda,$

$d,$

$q,$

$\alpha,$$C_{0},$$d,$$\lambda$$C_{0}$$c,$$\lambda,$$\beta$$C_{T,y}$$C_{T,x}$$\Vert v\Vert_{X_{\lambda}}$$\beta’$$\Vert v\Vert_{X_{\lambda}}$$C_{0},$$\alpha$$\Vert v\Vert_{Y_{T,x}^{q,\lambda}}$$q,$$(or \Vert v\Vert_{Y_{T,y}^{q,\lambda}})_{f}$$\Vert v\Vert_{X_{\lambda}}$Remark 1.3 In the proof of Theorem 1.2 we will also show that$(P_{K,v}f)(\cdot, s, \cdot)\in C([s, \infty);L^{p}(\mathbb{R}^{d}))$if$f\in L^{p}(\mathbb{R}^{d})$,$1\leq p<\infty$(1.28),and the energy inequality$\Vert\theta(t)\Vert_{L^{2}}^{2}+2\int_{s}^{t}\mathcal{E}_{K}^{(\tau)}(\theta(\tau), \theta(\tau))d\tau\leq\Vert f\Vert_{L^{2}}^{2}$

for $t>s\geq$ Oif$f\in L^{2}(\mathbb{R}^{d})$(1.29),.

The estimates (1.26) and (1.27) show the H\"older continuity of the fundamental solution, where the H\"older exponents and the constant C’ are estimated uniformly in time andincrease, if grows
andcan be larger as
space, while the constants
andat $|x|arrow\infty$ . We note that for some class of $(K, v)$ and solutions the H\"older continuity
is obtained in [5, 14] for the critical case and also in [8, 21] for the supercritical case. In$C_{T,x}$$|x|$$C_{T,y}$

$|y|$

$v$

was treated under[5, 14, 8] the case
and
but
(1.7)
the
and
(C2),
[21]
with
case
dealt
the condition
andwithout (C2).
In order to prove Theorem 1.2 it is important to obtain the a priori estimates for
fundamental solutions of the approximate equations which are, roughly speaking, of the
, and
and
with $\delta>0$ . Hereform
$K$
are suitable mollifications of and . It is more or less well known that the uniqueexistence of fundamental solutions holds for such mollified equations due to the fact that$v\in L^{\infty}(0, \infty;(\mathcal{L}^{2d/\alpha,2d/\alpha-d})^{d})$$A_{K}(t)=(-\triangle)^{\alpha/2}$$v\in(C^{1-\alpha}((0, \infty)\cross \mathbb{R}^{d}))^{d}$$\partial_{t}\theta+\delta(-\triangle)^{\tilde{\alpha}/2}\theta+A_{K^{-}}(t)\theta+\tilde{v}\cdot\nabla\theta=0$$\tilde{v}$$v$$\tilde{\alpha}\in(1,2)$$\tilde{K}$49
the leading term is the extra diffusion term. So our main step is to prove the a priori(equi-)continuity estimates of the fundamental solutions. For the purpose we will use
the Nash-type arguments by Komatsu [16, 17] where he studied the non-local diffusion
equations without the drift term. As in [18, 16, 17], the arguments consist of four steps;
the moment bound, the relative entropy bound, the overlap estimate, and the iteration
estimate. However, due to the presence of the nonsmooth drift term, it seems to bedifficult to obtain these estimates. In order to overcome the difficulty, we will derivethese estimates in time-dependent coordinates along the trajectory determined by a local
average of , instead of the usual coordinates . Although similar coordinates were usedin [5, 8, 14], we have to choose the appropriate trajectory in each step carefully. We note
that, in fact, the arguments in [18, 16, 17] highly rely on the scaling property of (1.1), while
the above approximation does not preserve such property. As a result, for example, it isdifficult to get the equi-continuity estimates for solutions to the approximate equations,which causes another technicality in taking the limit and showing the desired estimate
rigorously.
Theorem 1.2 has an application for the global regularity of solutions to (QG) in thecritical case, as in [15, 5, 14]. Indeed, our result gives the alternative approach to thisproblem, based on the Nash-type arguments for fundamental solutions. In particular,
it should be noted that different from [5] we need not study extension problems to use
special property of the fractional Laplacian.
$v$$\mathbb{R}^{d}$References[1] M. Barlow, R. F. Bass, Z. Q. Chen, and M. Kassmann, Non-local Dirichlet forms
and symmetric jump processes, Trans. Amer. Math. Soc. 361 (2009) 1963-1999.

[2] R. F. Bass, M. Kassmann, and T. Kumagai, Symmetric jump processes: localization,
heat kernels and convergence, Ann. Inst. H. Poincar\’e Probab. Stat. 46 (2010) 59-71.
[3] R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal.
17 (2002) 375-388.

[4] K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian
perturbed by gradient operators, Commun. Math. Phys. 271 (2007) 179-198.

[5] L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion
and the quasi-geostrophic equation, Ann. of Math. (2) 171 (2010) 1903-1930.
[6] E. A. Carlen, S. Kusuoka, and D. W. Stroock, Upper bounds for symmetric Markov
transition functions, Ann. Inst. H. Poincar\’e Probab. Stat. 23 (1987) 245-287.

[7] P. Constantin, A. Majda, and E. Tabak, Formation of strong fronts in the 2-D
quasigeostrophic thermal active scalar, Nonlinearity 7 (1994) 1495-1533.

50[8] P. Constantin and J. Wu, H\"older continuity of solutions of supercritical dissipativehydrodynamic transport equations, Ann. Inst. H. Poincar\’e Anal. Non Lin\’eaire 26(2009) 159-180.
[9] M. Giaquinta, Multiple integrals in the calculus of vanations and nonlinear elliptic
systems, Princeton University Press, Princeton, NJ, 1983.

[10] L. Grafakos, Classical and modem Founer analysis, Pearson Education, Inc., Upper
Saddle River, NJ.

[11] T. Hmidi and S. Keraani, Global solutions of the super-critica12D quasi-geostrophic
equation in Besov spaces, Adv. Math. 214 (2007) 618-638.
[12] T. Jakubowski and K. Szczypkowski, Time-dependent gradient perturbations offractional Laplacian, J. Evol. Eq. 10 (2010) 319-339.

[13] M. Kassmann, A priori estimates for integro-differential operators with measurable
kernels, Calc. Var. Partial Differential Equations 34 (2009) 1-21.
[14] A. Kiselev, F. Nazarov, A variation on a theme of Caffarelli and Vasseur, Zap.
Nauchn. Sem. POMI370 (2010) 58-72.

[15] A. Kiselev, F. Nazarov, and A.Volberg, Global well-posedness for the critica12D
dissipative quasi-geostrophic equation, Invent. Math. 167 (2007) 445-453.

[16] T. Komatsu, Continuity estimates for solutions of parabolic equations associatedwith jump type Dirichlet forms, Osaka J. Math. 25 (1988) 697-728.
[17] T. Komatsu, Uniform estimates for fundamental solutions associated with non-local
Dirichlet forms, Osaka J. Math. 32 (1995) 833-860.
[18] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math.
80 (1958) 931-954.
[19] H. Osada, Diffusion processes with generators of generalized divergence form, J.
Math. Kyoto Univ. 27 (1987) 597-619.

[20] G. Seregin, L. Silvestre, V. \v{S}ver\’ak, and A. Zlatos, On divergence-free drifts, preprint,
arXiv: 1010.6025vl [math.AP].
[21] L. Silvestre, H\"older estimates for advection fractional-diffusion equations, preprint,
arXiv: 1009.5723v2 [math.AP].

[22] Qi S. Zhang, A strong regularity result for parabolic equations, Commun. Math.
Phys. 244 (2004) 245-260.