Global Schauder estimates for kinetic Kolmogorov-Fokker-Planck equations
-
Hongjie Dong
and
Timur Yastrzhembskiy
Abstract
We present global Schauder type estimates in all variables and unique solvability results in kinetic Hölder spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equations. The leading coefficients are Hölder continuous in the x, v variables and are merely measurable in the temporal variable. Our proof is inspired by Campanato’s approach to Schauder estimates and does not rely on the estimates of the fundamental solution of the KFP operator.
1 Introduction and main result
Let d ≥ 1,
where
The above equation appears in kinetic theory, theory of diffusion processes, and mathematical finance (see [1] and the references therein). In particular, (1.1) with −v ⋅ D x u replaced with v ⋅ D x u can be viewed as a linearization of the Landau equation (see [2]), an important model of weakly coupled plasma. We also mention that P is the infinitesimal generator of the Langevin diffusion process (see [3]), so that the time-reversed version of (1.1) can be viewed as a backward Kolmogorov equation for the Langevin process.
It is a fundamental problem to establish the maximal regularity for the KFP equation in various functional spaces such as Hölder spaces (see [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]) and L p spaces (see [14], [15], [16], [17] and the references therein) that is analogous to the theory developed for nondegenerate equations (see, for example, [18], [19], [20]). Such results play a crucial role in the studies of the conditional regularity of the Landau equation (see [7]) and boundary value problem for the Landau equation with the specular reflection boundary condition (see [21], [22]).
The goal of the paper is to establish global Schauder estimates in t, x, v variables for Eq. (1.1) with time irregular leading coefficients (see Theorem 1.6). We also show how the constant on the right-hand side of the a priori estimate (1.10) depends on the lower eigenvalue bound δ (see Assumption 1.3). This is relevant to the linearization of the Landau equation near the global Maxwellian because for such an equation, one has
Hence, localizing to the velocity shell |v|∼ 2 n , we obtain the equation of type (1.1) with δ ∼ 2−3n (see the details in [21], [22]). Our method is inspired by Campanato’s approach and does not involve any estimates of the fundamental solution to the KFP equation. See the details in Section 1.4. Previously, we have used a kernel-free approach to prove estimates in certain weighted-mixed norm L p spaces for the KFP equation with rough leading coefficients (see [16], [17]).
Before we state the main result and review the relevant literature, we introduce some notation.
1.1 Notation
In this section, α ∈ (0, 1] is a number, and
The usual Hölder space. For an open set
and the norm
Anisotropic Hölder spaces. For α ∈ (0, 1] and an open set
Furthermore, for an open set of the form
we set
Furthermore, we say that
We stress that ∂ t u and v ⋅ D x u are understood in the sense of distributions. The norm in this space is defined as
where
Kinetic quasi-distance and kinetic Hölder spaces. We denote
Note that ρ satisfies all the properties of the quasi-distance except the symmetry. By
We introduce the kinetic Hölder seminorm
and the kinetic Hölder space
equipped with the norm
Furthermore, for an open set of the form (1.3), we define
is finite.
Remark 1.1.
Due to Lemma B.1 (i), replacing ρ(z, z′) with
Remark 1.2.
Our definition of the spaces
where
Convention. By N = N(…) and θ = θ(…), we denote constants depending only on the parameters inside the parentheses. These constants might change from line to line. Sometimes, when it is clear what parameters N and θ depend on, we omit them.
1.2 Main results
Assumption 1.3.
The function a = (a ij (z), i, j = 1, …, d) is measurable, and there exists some δ ∈ (0, 1) such that
Assumption 1.4.
The function a is of class
Assumption 1.5.
The functions b = (b 1(z), …, b d (z)) and c = c(z) are bounded measurable such that
for some L > 0.
Definition 1.1.
For
When s ∈ (0, 1/2), for any Lipschitz function
is valid.
Theorem 1.6.
Let α ∈ (0, 1), and Assumptions 1.3–1.5 be satisfied. Then, the following assertions hold.
For any
where
There exist numbers
such that for any
where N = N(d, α, K).
For any λ ≥ λ 0 (see the assertion (ii)) and
For any finite S < T and
has a unique solution
where
Remark 1.7.
In the case when a ij = a ij (t) and b ≡ 0, c ≡ 0, by a scaling argument (see Lemma 2.1), we conclude that (1.12) holds for any λ > 0. By using a compactness argument as in the proof of Theorem 1.6, one can show that the assertion (iii) of the above theorem is also valid for any λ > 0 in that case.
Corollary 1.8
(Kinetic interpolation inequalities). For any
where N = N(d, α).
It is easy to see that
Corollary 1.9
(‘Equivalence’ of
(ii) Let R > 0. If
Corollary 1.10
(Interior Schauder estimate). Let R > 0 and r ∈ (0, R) be constants. For any
where
1.3 Related works
In this section, we give a brief overview of the literature related to the Schauder estimates for the second-order nondegenerate parabolic equations and KFP equations.
Classical Schauder estimates. This theory asserts that if all the coefficients and the nonhomogeneous term are Hölder continuous with respect to all variables, then so are the second-order (spatial) derivatives of the solution. Such estimates can be proved either by using the integral representation of solutions and the bounds of the higher-order derivatives of the fundamental solution to the heat equation (see, for example, [18]) or by ‘kernel-free’ methods (see [23], [19], [24], [25], [26]).
Schauder estimates for parabolic equations with time irregular coefficients. In was showed in [31] that if for the nondegenerate parabolic equation, the coefficients and the nonhomogeneous term are of class
Schauder estimates for the KFP equations with Hölder continuous coefficients. A discussion of the Hölder theory and related results for the KFP equation can be found in [38]. The global (partial) parabolic Schauder estimate (cf. [31]) is established in [11] under the additional assumptions that the leading coefficients a ij are independent of time and have a limit at infinity (see also [39] and the references therein). In the case when the leading coefficients are Hölder continuous in t, x, and v, the interior Schauder estimate was proved in [6], [7], [12]. Later, the authors of [9] established the global Schauder estimate in the Hölder space
Schauder estimates for the KFP equation with irregular coefficients. The partial parabolic Schauder estimates similar to that of [31] were investigated in [4], [5], [8]. Their results can be stated in the following general way: under the assumptions 1.3–1.5, the
[8] is concerned with the interior estimate, which is applied to the well-posedness problem for the Landau equation with a ‘rough’ initial datum,
Recently, the authors of [10] established Hölder estimates for general Kolmogorov equations with time-measurable leading coefficients by a method based on derivative bounds of the fundamental solutions of Kolmogorov operators. Regarding the kinetic Kolmogorov-Fokker-Planck equation with vanishing initial data, the main result in [10] (see Theorem 2.7 therein) is similar to Theorem 1.6 of our paper, although it is stated somewhat differently. The main distinction lies in the definition of the solution space. Definition 2.3 in [10] requires additional regularity in the v variable (see Definition 1.4 therein) such as
We also mention the article [13], where the interior Schauder estimate for the operator (1.2) was derived under the assumption that the leading coefficients satisfy a Dini type condition. A few remarks in order.
1.4 Strategy of the proof
The main part of the argument is the proof of the a priori estimate (1.10) for a sufficiently regular function u (see Lemma 4.1). We remark that the
First, we show how our argument works in the case when the coefficients a
ij
depend only on the temporal variable. Our goal is to estimate the mean-oscillation of
Here
f = Pu, χ(t) = f(t, x 0 − (t − t 0)v 0, v 0),
ϕ is a suitable cutoff function,
ν ≥ 2 is a number, which we will choose later.
In the general case, we perturb the mean-oscillation estimates in Proposition 3.1 by using the method of frozen coefficients (see Lemma 4.1) and follow the above argument.
1.5 Additional notation and remarks. Geometric notation
Average. For a function f on
Functional spaces. For an open set
Kinetic Sobolev spaces. For p ∈ [1, ∞] and an open set
Local kinetic Sobolev spaces. By L
p;loc(G) we denote the set of all measurable functions u such that for any
Remark 1.11.
Here we give a couple of examples of functions belonging to the spaces
As pointed out in Section 1.3, even if u = u(x, v) is smooth in x and v, it might not be of class
Here is an example of a function of class
We have
Again, either estimating the
Remark 1.12.
It follows from the interpolation inequality in the usual Hölder space (see Lemma B.2) that if
and this is why the suprema of D
v
u and
Remark 1.13.
The completeness of
Remark 1.14.
It is easy to see that the following product rule inequality holds:
where
1.6 Organization of the paper
In Section 2, we prove some auxiliary results including the equivalence of the kinetic Hölder and Campanato seminorms. In Section 3, we establish the mean-oscillation estimates of
Acknowledgements. The initial draft of this manuscript was completed while the second author was a Prager Assistant Professor of Applied Mathematics at Brown University. The author expresses gratitude to the Division of Applied Mathematics for providing an excellent working environment during his time there. In addition, both authors would like to thank Weinan Wang and Andrea Pascucci for drawing their attention to the articles [8], [10], respectively. Finally, the authors express their sincere gratitude to the anonymous referees for their interest and comments, which have improved the presentation of the manuscript.
2 Auxiliary result
Lemma 2.1.
Let p ∈ [1, ∞] and
Then,
We introduce a kinetic Campanato type seminorm
(cf. Chapter 5 in [23]).
Here is a version of Campanato’s result (cf. Theorem 5.5 in [23]).
Lemma 2.2.
Let α ∈ (0, 1] and
Then, one has
where N = N(d, α).
Proof.
The second estimate follows from the definitions of the seminorms. The proof of the first bound is split into three steps.
Step 1:
replacing ρ with its symmetrization
Assuming (2.5), by Lemma B.1 (ii), we only need to demonstrate that
Indeed, by Lemma B.1 (iv),
By the doubling property (see Lemma B.1 (v)) and Lemma B.1 (iv),
Hence, the left-hand side of (2.6) is dominated by
Next, we will consider the case T < ∞ and assume that T = 0, for the sake of simplicity. Note that if t 0 < − r 2, one has
If t 0 ≥−r 2, then,
Thus,
so that (2.6) holds.
Step 2:
estimate of the deviation of u from its average. In the remaining steps, we follow the argument of Theorem 5.5 in [23] closely. Here we prove that for a.e.
First, let r
n
= 2−n
r and denote
To prove this, we note that for any Lebesgue measurable sets of finite measure A ⊂ A′,
This combined with the doubling property (see Lemma B.1 (v)) and (2.6) yields (2.9). Then, by using telescoping series and (2.9), we obtain
Furthermore, by the Lebesgue differentiation theorem in spaces of homogeneous type (see Lemma 7 in [42]) and Lemma B.1 (v),
Then, passing to the limit in (2.11) as n → ∞, we prove (2.8).
Step 3:
proof of (2.5). We fix any two points
By (2.8), we have
Next, to estimate J 3, we use an argument similar to that of (2.7). By Lemma B.1, (2.10), and the doubling property (Lemma B.1 (v)), we obtain
Combining (2.12)–(2.14), we prove (2.5) for a.e.
3 Estimate for the model equation
In this section, we assume that the coefficients a ij are independent of x, v and satisfy Assumption 1.3. We denote
Our goal is to prove a mean-oscillation estimate for
Proposition 3.1.
Let ν ≥ 2, α ∈ (0, 1), r > 0 be numbers,
Definition 3.1.
For −∞ ≤ T
1 < T
2 ≤ ∞, we write
Lemma 3.2
(cf. Lemma 5.2 in [16]). Let
R ≥ 1 be a number,
- (3.4)
u satisfy P 0 u = f in
and, furthermore, there exists θ = θ(d) > 0 such that
Proof.
We may assume that the right-hand side of (3.6) is finite. Let ϕ
n
, n ≥ 1, be a sequence of
with N independent of n.
Note that
Then, by Lemma 5.2 in [16], one has
and
By (3.7), for any r > 0,
Then, by using this and (3.4), and passing to the limit as n → ∞ in (3.8), we prove (3.5).
Next, we prove the bound for
Finally, due to the last inequality and a duality argument, the left-hand side of (3.6) is bounded by the limit supremum of the right-hand side of (3.9) as n → ∞. Now (3.6) follows from the above, (3.10), and (3.4). □
The following ‘nonlocal’ lemma is similar to Lemma 5.5 of [16] and Lemma 3.8 in [17]. In the present authors’ opinion, such ‘nonlocal’ lemmas are the technical novelties of the papers [16], [17], and the current article.
Lemma 3.3.
Let
We have
For any r ∈ (0, 1),
where
Proof.
First, multiplying u by a suitable cutoff function ϕ = ϕ(t) and using Corollary A.3, we conclude that
Let u ɛ be the mollification of u in the x variable with the standard mollifier and note that
By the above facts, we conclude
(3.12)Consequently, by the interior S 2 estimate (see Lemma A.5), for any 0 < r < 1,
where N = N(d, δ, r). Passing to the limit as ɛ → 0 in the above inequality and in (3.12), we prove the assertion (i).
We inspect the argument of Lemma 5.5 in [16]. In the sequel, N = N(d, r). Let
where
and
Estimate of
Then, by Theorem A.2 and the fact that |a| ≤ δ −1, we have
Furthermore, by (3.12) and the interior S 2 estimate in Lemma A.5, the last term is bounded by
Finally, due to the L p -boundedness of the Riesz transform and the Hörmander-Mikhlin inequality, we have
Estimate of Comm. We denote
Hence, we have
By the explicit representation of
By the Minkowski inequality,
By the Cauchy-Schwartz inequality, for any z ∈ Q r ,
Then, by using Minkowski inequality again, we get
Finally, combining (3.13)–(3.15), we obtain (3.11) with u replaced with u ɛ . Passing to the limit as ɛ → 0, we prove (3.11). □
Lemma 3.4
(Lemma 5.6 (i) in [16]). Let
Lemma 3.5.
Let
Proof. Step 1:
L 2 estimate of derivatives. Here we will show that for j ∈ {0, 1} and l + m ≥ 1, and 1/2 ≤ r < R ≤ 1,
To do that, we follow the argument of Lemma 5.6 in [16]. By mollifying u in the x variable, we may assume that u is smooth as a function of x.
Case 1: j = 0 = l, m ≥ 1. We will show that for any m ≥ 1,
where N = N(d, r, R). We prove this inequality by induction. Obviously, the estimate holds for m = 1. Furthermore, for any multi-index α of order m ≥ 1, one has
By the interior S 2 estimate in Lemma A.5, for r < r 1 < R,
Note that the first term on the right-hand side of (3.20) is bounded by the right-hand side in the equality (3.18) by the induction hypothesis. To handle the second term, note that for any nonempty multi-index β,
Then, by Lemma 3.4, for some r 1 < r 2 < 1,
Thus, the inequality (3.18) is valid. To make this argument rigorous, one can use the method of finite difference quotients.
Case j = 0, l ≥ 1, m ≥ 0. Arguing as in (3.22) and using (3.21) and Lemma 3.4, we get
Case 3: j = 1, l + m ≥ 1. Note that the function
The above formula combined with (3.18) and (3.23) yields
Thus, (3.17) holds.
Step 2:
L ∞ estimate of derivatives. By (3.17) and the Sobolev embedding theorem, for any l, m ≥ 0 such that l + m ≥ 1,
To estimate
Step 3:
proof of (3.16). Observe that
Then, by (3.26) and the Poincaré inequality,
where j ∈ {0, 1} and either m ≥ 2 or l ≥ 1. Finally, we denote
and observe that
By the above identities, the desired estimate (3.16) follows from (3.27) with U 1 in place of u. □
Lemma 3.6.
Invoke the assumptions of Lemma 3.5 and assume, additionally, that
where N = N(d, j, l, m).
Proof.
The desired assertion follows from (3.16) in Lemma 3.5 and (3.11) in Lemma 3.3. □
Proof of Proposition 3.1.
We may assume that the series involving P 0 u in (3.2) converges. Denote f = P 0 u. We split u into the ‘caloric’ part and a remainder and estimate each of the terms (see Section 1.4). After that, we prove the desired bounds of I 1 and I 2.
‘Remainder’ term. Let
u 1 be the unique
(3.29)(see Definition A.1 and Theorem A.2 (iii)),
u 2 = u 2(t, v) be the unique solution in the usual parabolic Sobolev space
(3.30)
Next, we use a scaling argument. By
Furthermore, by Lemma 3.2, there exists some θ = θ(d) > 0 such that for any R ≥ 1,
Next, note that for any ϰ, c > 0 and
Combining (3.31)–(3.32) with the above identity, we obtain for any R ≥ 1,
where
‘Caloric’ term. Denote
Note that
the conditions of Lemma 3.6 are satisfied due to (3.35) and the facts that
Furthermore, by (3.35) and Lemma 3.3 (i), we have
is valid. Hence, by Lemma 3.4,
Combining (3.36)–(3.37) with the identity
we obtain
Estimate of I 1 . First, note that by (3.34) with R = 1,
This combined with (3.39) and the triangle inequality give the desired estimate:
Estimate of I 2 . By (3.33) with R = 1,
and hence, by the triangle inequality, we only need to estimate I 2 with u replaced with u c .
Next, by using (3.38), we get
where
The term J 1 is estimated in (3.33) with R = 1. Furthermore, using (3.34) with R = 2 k gives
Noticing that F l (2 k ) = F l+k (1) and changing the index of summation k → k + l, we obtain
Combining the inequalities (3.40)–(3.41), (3.33), we prove the estimate of I 2 in (3.3) with u replaced with u c . As was mentioned above, this implies the desired bound of I 2. □
4 Proof of Theorem 1.6
In this section, we first show a few intermediate results and then prove Theorem 1.6.
Lemma 4.1.
For any α ∈ (0, 1) and
where N = N(d, α, K) > 0 and θ = θ(d, α) > 0.
Proof.
The idea is to perturb the mean-oscillation estimates in Proposition 3.1 to bound the Campanato’s seminorms (see (2.3)) of
Step 1:
freezing the coefficients. We fix some
By the identity
and Proposition 3.1 with a replaced with
where N = N(d), and
Next, by Lemma B.4 (i) and Assumption 1.4,
Step 2:
Campanato type argument. Estimate of
Note that ψ
1 is a nondecreasing function bounded by
where θ = θ(d) > 0,
Let
By a standard iteration argument (cf. Lemma 5.13 of [23]), we get
The latter combined with Lemma 2.2 yields
Estimate of
Then, multiplying (4.3) by |Q r |1/2 and using the above inequality combined with (4.4)–(4.5), we get
As above, we conclude that
Adding the last inequality to (4.7) gives
By using the interpolation inequality in Remark 1.12, we may replace B with
Lemma 4.2.
For any α ∈ (0, 1), there exists λ
0 as in (1.11) such that for any λ ≥ λ
0 and
Proof. Step 1:
case when
b ≡ 0
,
c ≡ 0
. We use S. Agmon’s method to derive (1.12) from (4.1). In particular, by this method, we are able to prove the bounds of
Agmon’s method (cf. Lemma 6.3.8 in [20]). Denote
Let ζ be a smooth cutoff function on
We choose such
We will extract the estimates of u and D v u from the above identities.
Estimate of u, D
v
u. By the product rule inequality in Remark 1.14, for any
where X is either
and a similar bound holds with sine instead of cosine, where N 1 = N 1(α, ζ). Combining (4.8)–(4.11) gives
where N = N(d, α).
Estimate of
where
By (4.11), (4.13)–(4.14), and (4.10), for λ, k ≥ 1,
where N = N(d, α) > 0.
Combining (4.12) with (4.15) and sending k → ∞, we get
By taking
Estimates of the transport term. By the identity
and Assumptions 1.3–1.4, and the product rule inequality, we get
and the right-hand is bounded by that of (1.12). Similarly, we can bound the L ∞ norm of the transport term.
Estimates of
Next, we claim that
If (4.17) is true, the term
is bounded (see, for example Theorem 1.3 in [43]) and a scaling argument, we conclude that
is also bounded by the right-hand side of (1.12).
To prove (4.17), we use a mollification argument. Let
By standard arguments, we have
which gives (4.17).
Step 2: adding the lower-order terms. By using (1.12) and the triangle inequality, we obtain (1.12) with the right-hand side replaced with
where
For sufficiently large λ ≥ λ 0 with λ 0 as in (1.11), the terms on the right-hand side of (4.20) can be absorbed into the left-hand side of (1.12). □
Proof of Theorem 1.6.
We prove the assertions in the following order: (iii), (ii), (iv), and (i). In particular, we will see that (ii) is an immediate corollary of (iii).
Proof of (iii) and (ii)
. Uniqueness. We only need to show that in the case when f ≡ 0, any solution u of class
Then by Lemma 4.2 and the product rule inequality in Remark 1.14, for any λ ≥ λ 0,
where ‖ ⋅‖ is the
Existence. Proof by a compactness argument. Let
Note that a n , b n , c n , f n satisfy the assumptions of Corollary A.4, and furthermore, by the product rule inequality (see Remark 1.14),
Hence, by Corollary A.4, the equation
has a unique solution
where [ ⋅ ] is the
Using the Arzela-Ascoli theorem and Cantor’s diagonal argument, from (4.21) we conclude that there exists
Proof of (iv) . The assertion is derived in a standard way by using (ii) and an exponential weight in the temporal variable.
Proof of (i)
. Note that (1.10) does not follow from (1.12) by setting λ = λ
0 in (1.12). Indeed, the latter gives an estimate weaker than (1.10) since it has extra terms involving
Step 1. We claim that Proposition 3.1 still holds if
Note that
We need to show that Lemma 3.3 still holds for
Step 2: proof of (1.10)
. The argument is the same as that of Lemma 4.1 with one modification: we do not need to use an iteration argument to conclude that
5 Proof of Corollaries 1.8–1.10
Proof of Corollary 1.8.
By a scaling argument, it suffices to prove the estimate in the case when ɛ = 1.
By (1.12) with a ij = δ ij , b = 0, c = 0, and λ = 1 (see Remark 1.7), we have
where ‖ ⋅‖ stands for the
By interpolating between
By using translation, it suffices to estimate
To this end, we will use a mollification argument (cf. (4.17)–(4.19)). Let
Estimate of J 3. Integrating by parts and using
Estimate of J i , i = 1, 2. By the fundamental theorem of calculus,
(5.1)where η i (v) = v i η(v). We note that since η is radial, one has
(5.2)Gathering the above estimates gives
Setting ɛ = |x 1 − x 2|1/3 in the above inequality, we obtain the desired estimate (1.16).
By translation and scaling (see Lemma 2.2), we only need to estimate
where t 1 = 1, t 2 = 0. As in the proof of the assertion (iii), we will use a mollification argument. The integrals J k , k = 1, 2, 3, need to be modified as follows:
Next, repeating the argument in (5.1)–(5.2), we get
For
By using the
Furthermore, by the fundamental theorem of calculus, we get
By using the fact that
Combining these estimates, we conclude that (1.17) holds. □
Proof of Corollary 1.9.
(i) We denote f = (∂
t
− v ⋅ D
x
)u − Δ
v
u. Since
(ii) Let
where by ‖ ⋅‖ we mean the
Proof of Corollary 1.10.
The proof is standard (cf. Theorem 7.1.1 in [19]). Let
and note that ζ
n
is a smooth function such that ζ
n
= 1 on
Next, uζ n satisfies the identity
Then, by Theorem 1.6 (ii), for any λ ≥ λ 0,
where
and ‖ ⋅‖ is the
We now estimate the terms I k , k = 1 − 4. In the sequel, N = N(d, α, K, L, r, R). By the product rule inequality (cf. Remark 1.14),
Arguing as above and using Assumptions 1.3–1.5 give
We now denote
and we set
where β > 2 and λ 1 > 1 will be determined later. Note that 4β > max{αβ + 3 + α, α + (2 + α)β}. Combining (5.4)–(5.5) gives
By the standard interpolation inequality (see Lemma B.2), for ɛ ∈ (0, 1), we have
Furthermore, we take
where ɛ 0 ∈ (0, 1) will be determined later, so that
We multiply both sides of (5.6) by 2−6βn and sum over n ∈ {0, 1, 2, …}. Due to (5.7)–(5.8), we get
Taking λ 1 > 1 large so that
we may drop the last two terms on the right-hand side of (5.9). The desired assertion is proved. □
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2055244
Award Identifier / Grant number: DMS-2350129
Funding source: Simons Foundation
Award Identifier / Grant number: 007638
-
Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Conflict of interest: All other authors state no conflict of interest.
-
Research funding: H. Dong was partially supported by a Simons fellowship, grant no. 007638, the NSF under agreements DMS-2055244 and DMS-2350129.
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Data availability: Not applicable.
Appendix A. S 2 regularity results for the KFP equations
Definition A.1.
We say that
holds in
Theorem A.1
(see Theorem 2.6 of [16]). Let α ∈ (0, 1], a be a function satisfying Assumptions 1.3–1.4 and
Theorem A.2
(see Theorem 4.1 of [16]). Let a = a ij (t) be a function satisfying Assumption 1.3 and recall the notation (3.1). Then, the following assertions hold.
For any λ ≥ 0 and
where
has a unique solution
For any finite numbers S < T and
where
Corollary A.3.
For any
Proof.
Let
Corollary A.4.
Invoke the assumption of Theorem A.1 and assume, additionally, that
and
Proof.
To make the argument presented below rigorous, one needs to use the method of finite-difference quotients. By using an induction argument similar to that used in the proof of Lemma 3.5, one can show that for any multi-indexes α and β, one has
We multiply the above identity by U, integrate over
An application of the Sobolev embedding theorem finishes the proof of this assertion. □
Lemma A.5
(Interior S
2 estimate, see Lemma 4.5 in [16]). Let a = a(t) satisfy Assumption 1.3,
where P 0 is defined by (3.1).
Appendix B
Lemma B.1
(Lemma 3.1 in [16]). Let r > 0 be a number. Then, the following assertions hold.
For any
For any
The function
One has
where
For T ∈ (−∞, ∞],
so that the triple
For the proof of the following inequality see, for instance, Lemma 6.3.1 in [19].
Lemma B.2
(Standard interpolation inequality in Hölder spaces). Let Ω be either
where N = N(d, k, α, j, β, Ω). In the case when
Lemma B.3.
Let α ∈ (0, 1] and
Proof. We note that by using Lemma 2.1 and a scaling argument, we only need to prove the assertions with ɛ = 1. Furthermore, due to the presence of the
We then obtain
and, thus, (B.1) is valid. □
Lemma B.4.
Let α ∈ (0, 1), c ≥ 1, r > 0,
where N = N(α).
Proof. (i) Denote
Then, the series on the left-hand side of (B.2) is less then
and hence, (B.2) is true.
(ii) For any z
1, z
2 such that
The last inequality and the fact that
imply the validity of the assertion (ii). □
Lemma B.5.
Let s ∈ (0, 1/2).
For any Schwartz function u, the following pointwise formula holds:
This formula is also valid for
For any
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Articles in the same Issue
- Frontmatter
- Editorial
- Preface for the special issue in honor of Robert Fefferman
- Research Articles
- Sobolev extension in a simple case
- Soliton resolution and channels of energy
- Matrix weights and a maximal function with exponent 3/2
- An introduction to the distorted Fourier transform
- The Neumann function and the L p Neumann problem in chord-arc domains
- An example related to Whitney’s extension problem for L 2,p (R2) when 1 < p < 2
- Convergence rates of eigenvalue problems in perforated domains: the case of small volume
- The initial-value problem for a Gardner-type equation
- Global Schauder estimates for kinetic Kolmogorov-Fokker-Planck equations
- Maximal estimates for strong arithmetic means of Fourier series
- On the L p -boundedness of Calderón-Zygmund operators
- Current perspectives on the Halo Conjecture
- On certain trilinear oscillatory integral inequalities
Articles in the same Issue
- Frontmatter
- Editorial
- Preface for the special issue in honor of Robert Fefferman
- Research Articles
- Sobolev extension in a simple case
- Soliton resolution and channels of energy
- Matrix weights and a maximal function with exponent 3/2
- An introduction to the distorted Fourier transform
- The Neumann function and the L p Neumann problem in chord-arc domains
- An example related to Whitney’s extension problem for L 2,p (R2) when 1 < p < 2
- Convergence rates of eigenvalue problems in perforated domains: the case of small volume
- The initial-value problem for a Gardner-type equation
- Global Schauder estimates for kinetic Kolmogorov-Fokker-Planck equations
- Maximal estimates for strong arithmetic means of Fourier series
- On the L p -boundedness of Calderón-Zygmund operators
- Current perspectives on the Halo Conjecture
- On certain trilinear oscillatory integral inequalities
