In Chapter 2, we introduce and study the following Sobolev spaces Sobolev spaces over a shift-invariant Banach space: - Inhomogeneous Sobolev spaces and homogeneous Sobolev spaces over t
Some results of real harmonic analysis
Littlewood-Paley decomposition
We take an arbitrary functionϕin the Schwartz class S(R d )and whose Fourier trans- form ϕˆ is such that
We denote byS j and∆ j , respectively, the convolution operators withϕ j andψ j Finally, the set {S j ,∆ j }j∈ Z is the Littlewood-Paley decomposition, so that
To be more precise, we should say 'a decomposition', because there are dierent possible (equivalent) choices for the function ϕ On the other hand, for an arbitrary tempered
8 distribution f, the last identity gives f =S 0 f +X j≥0
The interest in breaking down a tempered distribution into a sum of dyadic blocks, denoted as \$\Delta_j f\$, which have their support in Fourier space localized within a corona, arises from the favorable properties of these blocks concerning differential operations This concept is exemplified by the well-known Bernstein's lemma in \$\mathbb{R}^d\$, with its proof available in [44].
Lemma 1.1.1 Let 1 ≤ p ≤ q ≤ ∞ and k ∈N, then one has sup
|α|=k k∂ α fk p ≃R k kfk p and kfk q R d( 1 p − 1 q ) kfk p whenever f is a tempered distribution in S 0 whose Fourier transform f(ξ) ˆ is supported in the corona |ξ| ≃ R
In the case of a function whose support is a ball (as, for instance, for S j f) the lemma reads as follows
Lemma 1.1.2 Let 1 ≤ p ≤ q ≤ ∞ and k ∈N, then one has sup
|α|=k k∂ α fkp R k kfkp and kfk q R d( 1 p − 1 q ) kfk p whenever f is a tempered distribution in S 0 whose Fourier transform f(ξ) ˆ is supported in the ball |ξ|.R.
The decomposition of unity equations, introduced by Littlewood and Paley in the early 1930s, plays a crucial role in estimating the \$L^p\$-norm of trigonometric Fourier series for \$1 < p < \infty\$ Unlike the trivial case of \$p = 2\$, one cannot guarantee that a generic Fourier series belongs to the \$L^p\$ space solely based on its Fourier coefficients However, this assurance holds true when considering its dyadic blocks For a function \$f\$ (which need not be periodic), the equivalence states that for \$1 < p < \infty\$, the relationship \$\|f\|_p \sim \|S_0 f\|_p\$ is established.
It is even easier to prove that the classical Sobolev spaces H s = H 2 s , s ∈ R can be characterized by the following equivalent norm kfk H s ≃ kS 0 fk 2 + X j≥0
The general norm \( kf k_{H^p_s} \) is defined as \( (I - \Delta)^{s/2} f \) for \( s \in \mathbb{R} \) and \( 1 < p < \infty \), which corresponds to the Sobolev-Bessel spaces \( H^p_s \) When \( s \) is an integer, these spaces reduce to the well-known Sobolev spaces \( W^p_s \), where the norms are expressed as \( kf k_{W^p_s} = P \).
|α|≤sk∂ α fk p we will see in the next section how the equation (1.4) has to be modied.
Before defining the Besov spaces essential for our analysis of the Navier-Stokes equations, we should revisit the homogeneous decomposition of unity This decomposition is similar to equation (1.1) but also incorporates all low frequencies, specifically those where \( j < 0 \).
If we apply this identity to an arbitrary tempered distribution f, we may be tempted to write f =X j∈ Z
The identity in equation (1.5) differs from equation (1.2) as it lacks meaning in the space \( S_0 \) for several reasons Firstly, the sum in equation (1.5) may not converge in \( S_0 \) For instance, if we take a test function \( g \in S \) with a Fourier transform equal to 1 near the origin, the quantity \( \langle \Delta_j f, g \rangle \) remains a positive constant independent of \( j \) Furthermore, even if the sum converges, this convergence must be interpreted modulo polynomials, particularly for specific functions \( P \).
∆ j P = 0 for all j ∈ Z A way to restore the convergence is to suciently derive the formal series P j∈ Z as it stated in the following lemma (see [4, 5, 55] for a simple proof).
Lemma 1.1.3 For any tempered distribution f there exists an integer m such that for any α,|α| ≥m the series P j2, be such that s d < 1 p < 1
(a) There exists a positive constant δ s,p,r,d such that for all T > 0 and for all u 0 ∈ S 0 (R d ) with div(u 0 ) = 0 satisfying
L r ([0,T ]; ˙ H p s ) ≤δ s,p,r,d , (2.25) there is a unique mild solution u ∈ L r [0, T ]; ˙ H p s ) for NSE.
, then the inequality (2.25) holds when T (u 0 ) is small enough.
(b) If 2 r + d p − s = 1 then there exists a positive constant δ s,p,d such that we can take T = ∞ whenever e ã∆ u 0 L r ([0,∞]; ˙ H p s )≤δs,p,d
Proof (a) From Lemma 2.2.5, we have the estimate
L r ([0,T ]; ˙ H p s ) ≤C s,p,r,d T 1 2 (1+s− 2 r − d p ) , where C s,p,r,d is a positive constant independent of T From Theorem 1.5.1 and the above inequality, we deduce the existence of a solution to the Navier-Stokes equations on the interval(0, T)with
If e ã∆ u 0 ∈L r ([0,1]; ˙H p s ) then this condition is fullled forT =T(u 0 ) small enough, this is obvious for the case when 2 r + d p −s < 1 since lim
2 r + d p −s= 1, the condition is fullled since we have lim
L r ([0,1]; ˙H p s ) From Lemma 1.1.8, if u0 ∈ S 0 (R d ) the two quantities u0
Solutions to the Navier-Stokes equations with initial value in the
Lemma 2.2.8 Let d ≥ 3 and 2 ≤ q ≤ d Then the bilinear operator B(u, v)(t) is continuous from
, and we have the inequality
, (2.26) where C is a positive constant and independent of T.
Proof Applying Lemma 2.2.6 withr = 4, p= 2d−q 2dq , and s= d q −1, we get
From (b) of Lemma 1.1.5, we have
Finally, the estimate (2.26) can be deduced from the inequality (2.27) and the imbedding (2.28)
Lemma 2.2.9 Let d ≥ 3 and 2 ≤ q ≤ d Then the bilinear operator B(u, v)(t) is continuous from
, and we have the inequality
, (2.29) where C is a positive constant and independent of T.
Proof To prove this lemma by duality (in thex-variable), let us consider an arbitrary test function h(x)∈ S(R d ) Similar to the proof of Lemma 2.2.6, we have
However the dual space of B˙ q 0,2 0 is B˙ q 0,2 , therefore we get
From (b) of Lemma 1.1.5 and the estimate (2.30), we have
Finally, the estimate (2.29) can be deduced from the inequality (2.31).
Proof (a) From Lemma 1.1.8, we have the estimates e ã∆ u 0
Applying (b), (c), and (d) of Lemma 1.1.5 in order to obtain
L q = ˙H q 0 ,→B˙ q 0,q ,→B˙ q 0,4 ,→B˙ 2dq/(2d−q) −1/2,4 (2.33) From the inequality (2.32) and the imbedding (2.33), we get e ã∆ u 0
H q d/q−1 (b) Similar to the proof of (a) we have e ã∆ u 0
Combining Theorem 1.5.1 with Lemmas 2.2.1, 2.2.5, 2.2.8, and 2.2.10 we obtain the following existence result.
Theorem 2.2.11 Let 3 ≤ d ≤ 4 and 2 ≤ q ≤ d There exists a positive constant δ q,d such that for all T > 0 and for all u 0 ∈ H ˙ q d/q−1 (R d ) with div(u 0 ) = 0 satisfying e ã∆ u0
NSE has a unique mild solution u ∈ L 4 [0, T ]; ˙ H 2dq/(2d−q) d/q−1
Denoting w=u−e ã∆ u 0 , then we have w∈L 4 [0, T]; ˙H 2dq/(2d−q) d/q−1
H q d/q−1 , in particular, for arbitrary u 0 ∈ H ˙ q d/q−1 (R d )the inequality (2.34)holds when T (u 0 ) is small enough; and there exists a positive constant σ q,d such that for all u 0 ˙
Proof By applying Lemma 2.2.5 with r = 4, p = 2d−q 2dq , s = d q − 1, and notice that
L 4 [0,T]; ˙ H 2dq/(2d−q) d/q−1 ≤C q,d , where C q,d is a positive constant independent of T From Theorem 1.5.1 and the above inequality, we deduce that for any u 0 ∈H˙ d q −1 q such that div(u0) = 0, e ã∆ u0
4C q,d , NSE has a mild solution u on the interval (0, T) so that u∈L 4 [0, T]; ˙H 2dq/(2d−q) d/q−1
From the Lemma 2.2.8 and (2.35), we have B(u, u) ∈ L ∞ [0, T]; ˙Hq d/q−1
From (2) of Lemma 2.2.1, we have e ã∆ u 0 ∈L ∞ [0, T]; ˙Hq d/q−1
From (b) of Lemma 2.2.10, we have e ã∆ u 0
As \( T \) approaches 0, the left-hand side of inequality (2.34) converges to 0 Consequently, for any arbitrary \( u_0 \in H^{\dot{d}}_{q-1} \), there exists a sufficiently small \( T(u_0) \) that satisfies inequality (2.34) Additionally, there exists a positive constant \( \sigma_{q,d} \) applicable for all \( u_0 \).
B d/q−3/2,4 2dq/(2d−q) ≤ σ q,d and T =∞ the inequality (2.34) holds.
Remark 2.2.4 Theorem 2.2.11 in the particular case q = d is Proposition 20.1 in [46].
Theorem 2.2.12 Let 3 ≤ d ≤ 4 and 2 ≤ q ≤ d There exists a positive constant δ q,d such that for all T > 0 and for all u 0 ∈ H q d q −1 (R d ) with div(u 0 ) = 0 satisfying e ã∆ u 0
NSE has a unique mild solution u ∈ L 4 [0, T ]; H 2dq/(2d−q) d/q−1
H q d/q−1 , in particular, for arbitrary u 0 ∈ H q d q −1 the inequality (2.36) holds when T (u 0 ) is small enough;
Proof The proof of Theorem 2.2.12 is similar to the one of Theorem 2.2.11, by combining Theorem 1.5.1 with Lemmas 2.2.1, 2.2.5, 2.2.9, and 2.2.10.
This section explores mild solutions to the Navier-Stokes equations with initial values in the critical spaces \( \dot{H}^{\frac{d}{q}-1}(\mathbb{R}^d) \) for dimensions \( d \geq 3 \) and \( 1 < q \leq d \) We analyze two distinct cases: \( 2 < q \leq d \) and \( 1 < q \leq 2 \).
Solutions to the Navier-Stokes equations with initial value in the
spaces H˙ d q −1 q (R d ) for d≥3 and 2< q ≤d Lemma 2.2.13 Let d ≥ 3 and 2 < q ≤ d Then for all p such that
, the bilinear operator B(u, v)(t) is continuous from
, and we have the inequality
, (2.37) where C is a positive constant independent of T.
Proof Applying Lemma 2.2.6 withr =p and s= 2+d−p p , we get
Applying (e), (d), and (h) of Lemma 1.1.5 in order to obtain
Therefore the estimate (2.37) is deduced from the inequality (2.38) and the imbedding (2.39).
Lemma 2.2.14 Let 2< q < p 0 and for all u 0 ∈ H ˙ q d/q−1 (R d ) with div(u 0 ) = 0 satisfying e ã∆ u 0
NSE has a unique mild solution u ∈ L p [0, T ]; ˙ H p 2+d−p p ∩ L ∞ [0, T ]; ˙ H q d/q−1 Denoting w=u−e ã∆ u 0 , then we have w∈L p [0, T]; ˙H
, in particular, for arbitrary u 0 ∈ H ˙ q d/q−1 the inequality (2.42) holds when T (u 0 ) is small enough; and there exists a positive constant σ q,p,d such that for all u 0 ˙
Proof The proof of Theorem 2.2.15 is similar to the one of Theorem 2.2.11, by combining Theorem 1.5.1 with Lemmas 2.2.1, 2.2.5 (for r=p, s= 2+d−p p ), 2.2.13, and 2.2.14.
Remark 2.2.5 The caseq=dwas treated by several authors, see for example [11, 17, 37].However their results are dierent from ours.
Solutions to the Navier-Stokes equations with initial value in the
Lemma 2.2.16 Let d ≥ 3 and 1 < q ≤ 2 Then the bilinear operator B (u, v)(t) is continuous from
, and we have the inequality
, where C is a positive constant independent of T.
Proof Applying Lemma 2.2.6 withr = 2q, p= d+1−q dq , and s= d+2−2q q , we get
1 ˜ p = 2 p − s d = 1 q, and from (a) of Lemma 1.1.5, we have
Lemma 2.2.17 Assume that u 0 ∈ H ˙ q d q −1 with d ≥ 3 and 1 < q ≤ 2 Then e ã∆ u 0
Proof By using (a), (c), and (d) of Lemma 1.1.5 in order to obtain
Applying Lemma 1.1.6 and from the imbedding (2.43) we have the estimates e ã∆ u 0
Theorem 2.2.18 Let d ≥ 3 and 1 < q ≤ 2 There exists a positive constant δ q,d such that for all T > 0 and for all u 0 ∈ H ˙ q d/q−1 (R d ) with div(u 0 ) = 0 satisfying e ã∆ u 0
NSE has a unique mild solution u ∈ L 2q [0, T ]; ˙ H d+2−2q dq q d+1−q
H q d/q−1 , in particular, for arbitrary u 0 ∈ H ˙ q d/q−1 (R d )the inequality (2.44) holds when T (u 0 ) is small enough; and there exists a positive constant σ q,d such that for all ku 0 k B ˙ (d+1)/q−2,2q dq/(d+1−q)
Proof The proof of Theorem 2.2.18 is similar to the one of Theorem 2.2.11, by combining Theorem 1.5.1 with Lemmas 2.2.1, 2.2.5 (for r = 2q, p = d+1−q dq , s = d+2−2q q ), 2.2.16, and 2.2.17.
Remark 2.2.6 The caseq = 2was treated by several authors, see for example [11, 35, 46].However their results are dierent from ours.
Conclusions
In this section, for d ≥ 3, s ≥ 0, p > 1,and r > 2 be such that s d < 1 p < 1 2 + 2d s and
2 r+ d p −s ≤1, we investigate mild solutions to NSE in the spacesL r [0, T]; ˙H p s (R d )
First we obtain the existence of local mild solutions with an arbitrary initial tempered distribution datum in the Besov spacesB s−
2 r ,r p In the case of critical indexes 2 r −s+ d p = 1, we obtain the existence of global mild solutions when the norm of the initial tempered distribution datum in the Besov space B˙ s−
In the context of the Navier-Stokes equations, we explore the construction of mild solutions in the spaces \(L^\infty([0, T]; \dot{H}^{d/q - 1}_q(\mathbb{R}^d))\) and \(L^\infty([0, T]; H^{d/q - 1}_q(\mathbb{R}^d))\) This discussion builds upon the specific case presented by Lemarie-Rieusset when \(s = 0\) We will introduce two distinct algorithms tailored for solving the Cauchy problem, particularly when the initial conditions are defined within the Sobolev spaces \(H^{\dot{d/q - 1}}_q(\mathbb{R}^d)\) or \(H^{d/q - 1}_q(\mathbb{R}^d)\).
We employ the rst algorithm to analyze scenarios where the initial data is in the spaces \( H^{\dot{d}}_{q-1}(\mathbb{R}^d) \) or \( H^{d}_{q-1}(\mathbb{R}^d) \) for dimensions \( 3 \leq d \leq 4 \) and \( 2 \leq q \leq d \) Our findings, particularly when \( q = d \), extend the results previously established in [46] Additionally, the second algorithm allows us to address cases where the initial data resides in critical spaces.
The study of the equation H˙ d q −1 q (R d ) is relevant for dimensions where \(d \geq 3\) and \(1 < q \leq d\) Notably, the specific cases of \(q = 2\) and \(q = d\) have been extensively analyzed by various researchers, as referenced in sources [11, 13, 16, 17, 34, 35, 37, 46, 57] Additionally, the Cauchy problem related to an incompressible magneto-hydrodynamics system, characterized by positive viscosity and magnetic resistivity, has been explored within the context of Besov spaces in reference [53].
Mild solutions in Sobolev spaces of negative order
Conclusions
This section introduces an alternative algorithm for developing mild solutions within the space \( L^\infty([0, T]; \dot{H}^s_p(\mathbb{R}^d)) \) for the Cauchy problem related to the Navier-Stokes equations (NSE) The initial conditions are assumed to reside in the Sobolev spaces \( \dot{H}^s_p(\mathbb{R}^d) \), where the parameters satisfy \( d \geq 2 \), \( p > \frac{d}{2} \), and \( d p - 1 \leq s < \frac{2p}{d} \).
Mild solutions with arbitrary initial values exist when \( T \) is sufficiently small Additionally, for any \( T > 0 \), mild solutions can be found when the initial value's norm is within the Besov spaces \( \dot{B}^{s-d} \).
In the context of Besov spaces, we establish a result that surpasses the findings of Cannone and Meyer, particularly when the conditions on the initial data are less stringent This is applicable when \( p > d \) and \( s = 0 \), with the parameters satisfying \( 1/q = 1/p - s/d \) being sufficiently small, and \( \tilde{q} > \max\{p, q\} \).
For critical indexes where \( p > d^2 \) and \( s = d p - 1 \), we can set \( T = \infty \) if the initial value's norm in the Besov spaces \( \dot{B}^{\tilde{q}-1,\infty}_{\tilde{q}}(\mathbb{R}^d) \) (with \( \tilde{q} > \max\{d, p\} \)) is sufficiently small This finding, applicable when \( s = 0 \) and \( p = d \), aligns with one of the Cannone theorems (refer to Theorem 3 in [15], p 195).
Mild solutions in the Sobolev-Fourier-Lorentz spaces
Sobolev-Fourier-Lozentz Space
Denition 2.4.1 (Fourier-Lebesgue spaces) (See [30].)
For 1 ≤ p ≤ ∞ , the Fourier-Lebesgue spaces L p (R d ) are dened as the space F −1 (L p 0 (R d )),
L p 0 ( R d ), where F and F −1 denote the Fourier transform and its inverse.
Denition 2.4.2 (Sobolev-Fourier-Lebesgue spaces).
For s ∈ R, and 1 ≤ p ≤ ∞ , the Sobolev-Fourier-Lebesgue spaces H ˙ L s p (R d ) are dened as the space Λ ˙ −s L p (R d ), equipped with the norm u ˙
Denition 2.4.3 (Fourier-Lorentz spaces) For 1 ≤ p, r ≤ ∞ , the Fourier-Lorentz spaces
L p,r (R d ) are dened as the space F −1 (L p 0 ,r (R d )),( p 1 0 + 1 p = 1), equipped with the norm f
Denition 2.4.4 (Sobolev-Fourier-Lorentz spaces).
For s ∈ R and 1 ≤ r, p ≤ ∞ , the Sobolev-Fourier-Lorentz spaces H ˙ L s p,r (R d ) are dened as the space Λ ˙ −s L p,r (R d ), equipped with the norm u ˙
Theorem 2.4.1 (Holder's inequality in Fourier-Lorentz spaces )
Let 1 < r, q, q < ˜ ∞ and 1 ≤ h, ˜ h, ˆ h ≤ +∞ satisfy the relations
Suppose that u ∈ L q, ˜ h and v ∈ L q, ˜ ˆ h Then uv ∈ L r,h and we have the inequality uv
Proof Let r 0 , q 0 , and q ˜ 0 be such that
It is easily checked that the following conditions are satised
L q, ˜ ˆ h (2.74) Now, the estimate (2.72) follows from the equality (2.73) and the inequality (2.74).
Theorem 2.4.2 (Young's inequality for convolution in Fourier-Lorentz spaces )
Let 1 < r, q, q < ˜ ∞ , and 1 ≤ h, ˜ h, ˆ h ≤ ∞ satisfy the relations
Suppose that u ∈ L q, ˜ h and v ∈ L q, ˜ ˆ h Then u ∗ v ∈ L r,h and the following inequality holds u∗v
Proof Letr 0 , q 0 , and q˜ 0 be such that
We can check that the following conditions are satised
1< r 0 , q 0 ,q˜ 0 δ ⊇ x:|1 B c n+1u 0 (x)|> δ , (2.83) and ∞ n=0∩ {x:|1 B c n u 0 (x)|> δ}=∅ (2.84) Note that
, where L d being the Lebesgue measure in R d We prove that
We have u ∗ 0 (t)≥δ for all t >0, from the denition of the Lorentz space, we get u 0
From (2.83), (2.84), and (2.85), we have n→∞limL d {x:|1 B n c u 0 (x)|> δ}
Fixt >0 For any >0, from (2.86) it follows that there existsn 0 =n 0 (t, ) large enough such that
≤t,∀n≥n 0 From this we deduce that u ∗ n (t)≤,∀n≥n 0 , therefore lim u ∗ n (t) = 0 n→∞
From (2.87) and (2.88), we apply Lebesgue's monotone convergence theorem to get n→∞lim
Now we return to prove Lemma 2.4.10 We prove that sup
Let p 0 and p˜ 0 be such that
The estimate (2.89) follows from the equality (2.90) and the estimate (2.91).
We claim now that limt→0t α 2 e t∆ u0 ˙
From the equality (2.90), we have t α 2 e t∆ u 0 ˙
For any >0, applying Holder's inequality in the Lorentz spaces and using Lemma 2.4.11, we have t α 2 e −t|ξ| 2 1 B n c |ξ| d p −1 uˆ 0 (ξ)
2 (2.93) for large enough n Fix one of such n and applying Holder's inequality in the Lorentz spaces, we have t α 2 e −t|ξ| 2 1B n|ξ| d p −1 uˆ0(ξ)
2 (2.94) for small enough t =t(n)>0 From estimates (2.93) and (2.94), we have, t α 2 e t∆ u 0 ˙
This article focuses on the analysis of the bilinear operator \( B(u, v)(t) \) defined in the Sobolev-Fourier-Lorentz spaces Specifically, Lemma 2.4.12 states that for all \( p \) satisfying \( 1 < p \leq d \), certain conditions will be examined.
, (2.95) the bilinear operator B(u, v)(t) is continuous from K p ˜ d
[ d p ] ,∞,T into K p p,1,T and the following inequality holds
, (2.96) where C is a positive constant and independent of T.
Setting the tensorK(x) = {Kl,k,j(x)}, we can rewrite the equality (2.98) in the tensor form Λ˙ d p −1 e (t−τ)∆ P∇ ã u(τ)⊗v(τ)
Applying Theorem 2.4.2 for convolution in the Fourier-Lorentz spaces, we have Λ˙ d p −1 e (t−τ)∆ P∇ ã u(τ)⊗v(τ)
Note that from the inequality (2.95), we can check that r and q satisfy the relations
From the equalities (2.99) and (2.101), we obtain
From the estimates (2.100), (2.102), and (2.103), we deduce that Λ˙ d p −1 e (t−τ)∆ P∇ ã u(τ)⊗v(τ)
− d ˜ p , this gives the desired result
Let us now check the validity of the condition (2.82) for the bilinear term B(u, v)(t). Indeed, from (2.104) limt→0
The estimate (2.96) is deduced from the inequality (2.104).
Lemma 2.4.13 Let 1 < p ≤ d Then for all p ˜ be such that
, (2.105) the bilinear operator B(u, v)(t) is continuous from K p ˜ d
[ d p ] ,1,T and the following inequality holds
, (2.106) where C is a positive constant and independent of T.
Proof First, arguing as in Lemma 2.4.12, we derive Λ˙ [ d p ]−1 e (t−τ)∆ P∇ ã u(τ)⊗v(τ)
Applying Theorem 2.4.2 for the convolution in the Fourier-Lorentz spaces, we have Λ˙ [ d p ]−1 e (t−τ )∆ P∇ ã u(τ)⊗v(τ)
Note that from the inequality (2.105), we can check that r and q satisfy the relations
From the equalities (2.107) and (2.109), we obtain
From the estimates (2.108), (2.110), and (2.111), we deduce that Λ˙ [ d p ]−1 e (t−τ)∆ P∇ ã u(τ)⊗v(τ)
=hd p i− d ˜ p , this gives the desired result
Now we check the validity of the condition (2.81) for the bilinear term B(u, v)(t) From (2.112) we infer that limt→0t α 2
Finally, the estimate (2.106) can be deduced from the inequality (2.112).
Theorem 2.4.14 Let 1 < p ≤ d and 1 ≤ r < ∞ Then for all p ˜ be such that
2d o , there exists a positive constant δ p,˜ p,d such that for all T > 0 and for all u 0 ∈ H ˙ L d p p,r −1 (R d ) with div(u 0 ) = 0 satisfying sup
NSE has a unique mild solution u ∈ K p ˜ d
In particular, the inequality (2.113) holds for arbitrary u 0 ∈ H ˙ L d p p,r −1 (R d )when T (u 0 ) is small enough, and there exists a positive constant σ p,˜ p,d such that we can take T = ∞ whenever u 0 ˙
Proof From Lemmas 2.4.9 and 2.4.13, the bilinear operator B(u, v)(t)is continuous from
[ d p ] ,1,T and we have the inequality
, where C p,˜ p,d is positive constant independent of T From Theorem 1.5.1 and the above inequality, we deduce that for any u0 ∈H˙ d p −1
≤ 1 4C p,˜ p,d , the NSE has a solution u on the interval (0, T) so that u∈ K p ˜ d
From Lemmas 2.4.9 and 2.4.12, and (2.114), we have
. From Lemma 2.4.8, we also have e ã∆ u 0 ∈L ∞ [0, T]; ˙H d p −1
L p,r , applying Theorem 2.4.3, we deduce that u 0 ∈H˙ [ d p ]−1
From (2.115), applying Lemma 2.4.10, we get e ã∆ u 0 ∈ K p ˜ d
K p p,r,T ˜ , we deduce that the left-hand side of the inequality (2.113) converges to 0 when T tends to 0 Therefore the inequality (2.113) holds for arbitrary u 0 ∈ H˙ d p −1
L p,r when T(u 0 ) is small enough Applying Lemmas 2.4.10 and 2.4.13, we conclude thatu∈ K p ˜ d
[ d p ] ,1,T. Next, applying Theorem 5.4 ([46], p 45), we deduce that the two quantities u 0 ˙
L p,∞ ˜ are equivalent, then there exists a positive constant σ p,˜ p,d such that T =∞ and (2.113) holds whenever u 0 ˙
Remark 2.4.5 From Theorem 2.4.3 and the proof of Lemma 2.4.10, and Theorem 5.4 ([46], p 45), we have the following imbedding maps
On the other hand, a function in H˙ d p −1
L p,r (R d )can be arbitrarily large in theH˙ d p −1
L p,r (R d )norm but small in the B˙ d ˜ p −1,∞
Solutions to the Navier-Stokes equations with the initial value in the
Lemma 2.4.15 Suppose that u 0 ∈ H ˙ L d p p,r −1 with d ≤ p < ∞ and 1 ≤ r < ∞ Then e ã∆ u 0 ∈ K p d,1,∞ ˜ for all p > p ˜
Proof We prove that sup
, where α=α(d,p) = 1˜ −d ˜ p. Let p 0 and p˜ 0 be such that
L p ξ ˜ 0 ,1 Applying Holder's inequality in the Lorentz spaces to obtain e −t|ξ| 2 |ξ| 1− d p |ξ| d p −1 uˆ 0 (ξ)
Therefore this gives the desired result e t∆ u 0
We claim now that limt→0t α 2 e t∆ u 0
For any >0, applying Lemma 2.4.11 and from the above proof we deduce that t α 2 e t∆ u 0
< for large enough n and small enought=t(n)>0.
Then the bilinear operator B(u, v)(t) is continuous from K p d,∞,T ˜ × K p d,∞,T ˜ into K p p,1,T , and we have the inequality
K p d,∞,T ˜ , (2.117) where C is a positive constant and independent of T.
Proof First, arguing as in Lemma 2.4.12, we derive Λ˙ d p −1 e (t−τ)∆ P∇ ã u(τ)⊗v(τ)
, where the tensor K(x) ={K l,k,j (x)} is given by the formula
Applying Theorem 2.4.2 for the convolution in the Fourier-Lorentz spaces, we have Λ˙ d p −1 e (t−τ)∆ P∇ ã u(τ)⊗v(τ)
Note that from the inequality (2.116), we can check that 1 < r < ∞ Applying Theorem 2.4.1, we have u(τ)⊗v(τ)
From the equalities (2.118) and (2.120) it follows that
From the estimates (2.119), (2.121), and (2.122), we deduce that e (t−τ)∆ P∇ ã u(τ)⊗v(τ) ˙
L p,∞ ˜ , where α=α(d,p) = 1˜ −d ˜ p. This gives the desired result
From (2.123) it follows the validity of (2.82) since limt→0
The estimate (2.117) can be deduced from the inequality (2.123).
Lemma 2.4.17 Let p > d ˜ , then the bilinear operator B(u, v)(t) is continuous from
K p d,∞,T ˜ × K p d,∞,T ˜ into K p d,1,T ˜ , and we have the inequality
, (2.124) where C is a positive constant and independent of T.
Proof First, arguing as in Lemma 2.4.12, we derive e (t−τ)∆ P∇ ã u(τ)⊗v(τ)
, where the tensor K(x) ={K l,k,j (x)} is given by the formula (2.107).
Applying Theorem 2.4.2 for the convolution in the Fourier-Lorentz spaces, we have e (t−τ)∆ P∇ ã u(τ)⊗v(τ)
From the equalities (2.107) and (2.126) it follows that
L r 0 ,1 ≃(t−τ) d 2 (1− 1 p ˜ ) (2.128) From the estimates (2.125), (2.127), and (2.128), we deduce that e (t−τ)∆ P∇ ã u(τ)⊗v(τ)
L p,∞ ˜ , where α=α(d,p) = 1˜ −d ˜ p. This gives the desired result
From (2.129) it follows the validity of (2.81) since limt→0t α 2
Finally, the estimate (2.124) can be deduced from the inequality (2.129).
The following lemma is a generalization of Lemma 2.4.17.
Lemma 2.4.18 Let d < p ˜ 1 < ∞ and d ≤ p ˜ 2 < ∞ be such that one of the following conditions d 0 and for all u 0 ∈ H ˙ L d p p,r −1 (R d ), with div(u 0 ) = 0 satisfying sup
NSE has a unique mild solution u ∈ ∩ q>p K q d,1,T ∩L ∞ ([0, T]; ˙H d p −1
In particular, the inequality (2.131) holds for arbitrary u 0 ∈ H ˙ L d p p,r −1 (R d ) with T (u 0 ) is small enough, and there exists a positive constant σ p,d ˜ such that we can take T = ∞ whenever u 0 ˙
Proof The proof of Theorem 2.4.19 is similar to the one of Theorem 2.3.5 by combining Lemmas 2.4.8,2.4.15,2.4.16, 2.4.18, and Theorem 1.5.1.
Remark 2.4.6 From the proof of Lemma 2.4.15 and Theorem 5.4 ([46], p 45), we have the following imbedding maps
On the other hand, a function in H˙ d p −1
L p,r (R d )can be arbitrarily large in theH˙ d p −1
L p,r (R d )norm but small in the B˙ d ˜ p −1,∞
Solutions to the Navier-Stokes equations with the initial value in the
We dene an auxiliary spaceK s,r,T which consists of functions u(t, x)such that u
In the cases=d−1, it is also convenient to dene the space Kd−1,r,T as the natural space
L ∞ [0, T]; ˙H L d−1 1,r with the additional condition that its elements u(t, x)satisfy limt→0 u(t, x) ˙
Lemma 2.4.20 Let 1 ≤ r ≤ r ˜ ≤ ∞ Then we have the following imbeddings
Proof It is deduced from Lemma 2.4.4 (a) and the denition ofKs,r,T.
Lemma 2.4.21 Suppose that u 0 ∈ H ˙ L d−1 1,r (R d ) with 1 ≤ r < ∞ Then e ã∆ u 0 ∈ K s,r,∞ with d−1< s < d.
Proof We prove that sup
We claim now that limt→0t α 2 e t∆ u 0 ˙
From the inequality (2.135), we have t α 2 e t∆ u 0 ˙
L ∞,r ξ For any >0, applying Lemma 2.4.11, we have t α 2
2, (2.136) for large enough n Fixed one of suchn, we have the following estimates t α 2
2 (2.137) for small enough t =t(n)>0 From the estimates (2.136) and (2.137), we have, t α 2 e t∆ u 0 ˙
Lemma 2.4.22 Let d − 1 < s < d Then the bilinear operator B (u, v)(t) is continuous from K s,∞,T × K s,∞,T into K s,1,T and we have the inequality
K s,1,T ≤C u K s,∞,T v K s,∞,T, (2.138) where C is a positive constant and independent of T.
Proof Using the Fourier transform we get
|ξ| 2s−d From the above estimates and Lemma 2.4.4 (b), we have
|ξ| 2s−d , this gives the desired result
Let us now check the validity of the condition (2.132) for the bilinear term B(u, v)(t). Indeed, from (2.139) limt→0t α 2
The estimate (2.138) is deduced from the inequality (2.139).
Lemma 2.4.23 Let d − 1 < s < d Then the bilinear operator B (u, v)(t) is continuous from K s,∞,T × K s,∞,T into K d−1,1,T and we have the inequality
K d−1,1,T ≤C u K s,∞,T v K s,∞,T, (2.140) where C is a positive constant and independent of T.
Proof First, arguing as in Lemma 2.4.22, we have the following estimates
L 1,∞ dτ , this gives the desired result
From (2.141) it follows (2.133) since limt→0
The estimate (2.140) can be deduced from the inequality (2.141).
Theorem 2.4.24 Let d − 1< s < dand 1 ≤ r < ∞ Then there exists a positive constant δ s,d such that for all T > 0 and for all u 0 ∈ H ˙ L d−1 1,r (R d ), withdiv(u 0 ) = 0 satisfying sup
NSE has a unique mild solution u ∈ K s,r,T ∩ L ∞ ([0, T ]; ˙ H L d−1 1,r ).
In particular, the inequality (2.142) holds for arbitrary u 0 ∈ H ˙ L d−1 1,r (R d )when T (u 0 ) is small enough, and there exists a positive constant σ s,d such that we can take T = ∞ whenever u0 ˙
The proof of Theorem 2.4.24 closely resembles that of Theorem 2.4.14 By utilizing Lemma 2.4.22 and Theorem 1.5.1, we establish the existence of a positive constant \$\delta_{s,d}\$ such that for any initial condition \$u_0 \in H^{\dot{L}}^{d-1}_{1,r}(\mathbb{R}^d)\$ with divergence \$\text{div}(u_0) = 0\$, the supremum is maintained.
NSE has a solution u ∈ Ks,∞,T Applying Lemmas 2.4.8 and 2.4.23 we deduce that u ∈
L ∞ ([0, T]; ˙H L d−1 1,r) Applying Lemma 2.4.21, we get e ã∆ u 0 ∈ K s,r,T From the denition of
K s,r,T , we deduce that the left-hand side of the inequality (2.142) converges to 0 when T tends to0 Therefore the inequality (2.142) holds for arbitrary u 0 ∈H˙ L d−1 1,r(R d )whenT(u 0 ) is small enough.
Next, from the inequality (2.134) with r=∞, we deduce that sup
, then there exists a positive constant σ s,d such that T = ∞ and (2.142) holds whenever u 0 ˙
Remark 2.4.7 The case r=∞was studied by Le Jan and Sznitman in [7] They showed that NSE are well-posed when the initial datum belongs to the spaceH˙ L d−1 1,∞ For1≤r 0, applying Theorem 1.1.10 (c) and note that (2.154), we have t α 2 − d 2 (4π) d/2 e − |ã|
2, (2.156) for large enough n Fix one of such n, applying Theorem 1.1.10 (a), we conclude that t α 2 − d 2
2, (2.157) for small enough t >0 From the estimates (2.155), (2.156), and (2.157) it follows that t α 2 e t∆ u 0 ˙
Finally, the embedding (2.145) is derived from the inequality (2.153), Lemmas 2.5.1 and 1.1.8.
Remark 2.5.3 In the case s = 0 and q = r = d, Lemma 2.5.4 is a generalization ofLemma 9in ([15], p 196).
On the continuity and regularity of the bilinear operator
In this subsection a particular attention will be devoted to the study of the bilinear operator B(u, v)(t) dened by (1.14) in Sobolev-Lorentz spaces.
Lemma 2.5.6 Let s, q ∈R be such that s≥0, q >1, and s d < 1 q ≤ s+ 1 d (2.158)
Then for all q ˜ satisfying s d < 1 ˜ q 0 and for all u 0 ∈ H ˙ L s q,r (R d ) with div(u 0 ) = 0 satisfying
NSE has a unique mild solution u ∈ K s, q,1,T q ˜ ∩L ∞ [0, T]; ˙H L s q,r
In particular, for arbitrary u 0 ∈ H˙ L s q,r with div(u 0 ) = 0 , there exists T (u 0 ) small enough such that the inequality
(b) If 1 < q ≤ d and s = d q − 1 then for any q ˜ be such that
2+ 1 2q − 1 2d,1 q o , there exists a positive constant σ q,˜ q,d such that if u 0 ˙
≤ σq,˜ q,d and T =∞ then the inequality (2.181) holds.
Proof From Lemmas 2.5.6 and 2.5.3, the bilinear operator B(u, v)(t) is continuous from
K s,˜ q,˜ q q,T × K q,˜ s,˜ q q,T into K s,˜ q,˜ q q,T and we have the inequality
, where C s,q,˜ q,d is a positive constant independent of T From Theorem 1.5.1 and the above inequality, we deduce the following: for anyu0 ∈H˙ L s q,r(R d )such that div(u 0 ) = 0, T 1 2 (1+s− d q ) sup
NSE has a mild solution u on the interval (0, T) so that u∈ K s, q,˜ q q,T ˜ (2.182)
Lemma 2.5.7 and the relation (2.182) imply that
On the other hand, from Lemma 2.5.2, we have e ã∆ u0 ∈L ∞
From Lemmas 2.5.4 and 2.5.6, we deduce that u∈ K s,˜ q,1,T q
Based on the definition of \( K_{q,r,T} \) and Lemma 2.5.4, we conclude that the left side of inequality (2.181) approaches 0 as \( T \) approaches 0 Consequently, inequality (2.181) is valid for any \( u_0 \in \dot{H}^L_{s,q,r}(\mathbb{R}^d) \) when \( T(u_0) \) is sufficiently small.
(b) From Lemma 1.1.8, the two quantities u 0
H ˙ d q −1 ˜ q are equivalent, then there exists a positive constant σ q,˜ q,d such that if u 0
≤σ q,˜ q,d and T =∞ then the inequality (2.181) holds.
Remark 2.5.5 In the case when the initial data belong to the critical Sobolev-Lorentz spacesH˙ d q −1
L q,r (R d ),(1< q ≤d, r ≥1), from Theorem 2.5.8 (b), we get the existence of global mild solutions in the spaces L ∞ ([0,∞); ˙H d q −1
L q,r (R d )) when the norm of the initial value in the Besov spaces B˙ d ˜ q −1,∞ ˜ q (R d ) is small enough Note that from Lemma 2.5.4 we have the following imbedding maps
This result is stronger than that of Cannone In particular, when q =r =d, s = 0, we get back the Cannone theorem (Theorem 1.1 in [12]).
Next, we consider the super-critical indexes s > d q −1.
Then for any q ˜ be such that
2+ s 2d,1 q o , there exists a positive constant δ s,q,˜ q,d such that for all T > 0 and for all u 0 ∈ H ˙ L s q,r (R d ) with div(u 0 ) = 0 satisfying
NSE has a unique mild solution u ∈ K s,˜ q,1,T q ∩L ∞ ([0, T]; ˙H L s q,r).
Proof Applying Lemma 1.1.8, the two quantities u 0
,the theorem is proved by applying the above inequality and Theorem 2.5.8.
Remark 2.5.6 In the case when the initial data belong to the Sobolev-Lorentz spaces
In the context of the spaces \( L^\infty([0, T]; \dot{H}^L_{s,q,r}(\mathbb{R}^d)) \) for any \( T > 0 \), we establish the existence of mild solutions when the initial value's norm in the Besov spaces \( B^{\dot{s} - (d/q - \tilde{q})}_{\infty, \tilde{q}}(\mathbb{R}^d) \) is sufficiently small, given the conditions \( q > 1 \), \( r \geq 1 \), \( s \geq 0 \), and \( d q - 1 < s < d q \) Additionally, we refer to the imbedding maps outlined in Lemma 2.5.4 for further insights.
By applying Theorem 2.5.9 with parameters \( q > d \), \( r = q \), and \( s = 0 \), we derive a proposition that surpasses the findings of Cannone and Meyer [11, 14] Notably, this result is achieved under significantly weaker conditions on the initial data.
Proposition 2.5.10 Let q > d Then for any q ˜ be such that q < q < ˜ 2q , there exists a positive constant δ q,˜ q,d such that for all T > 0 and for all u 0 ∈ L q (R d ) with div(u 0 ) = 0 satisfying
NSE has a unique mild solution u ∈ K 0,˜ q,1,T q ∩L ∞ ([0, T];L q ).
Remark 2.5.7 If in (2.183) we replace the B˙ d ˜ q − d q ,∞ ˜ q norm by the L q norm, then we get the assumption made in [11, 14] We show that the condition (2.183) is weaker than the condition in [11, 14] In Remark 2.5.6 we have showed that
However these two spaces are dierent Indeed, we have x
∈/ L q (R d ) On the other hand by using Lemma 1.1.8, we can easily prove that x
By applying Theorem 2.5.9 with parameters \( q = r = 2 \) and the condition \( d^2 - 1 < s < d^2 \), we derive a proposition that surpasses the findings of Chemin and Cannone Notably, this result is achieved under significantly weaker conditions on the initial data compared to their work.
Proposition 2.5.11 Let d 2 −1< s < d 2 Then for any q ˜ be such that
2, there exists a positive constant δ s,˜ q,d such that for all T > 0 and for all u 0 ∈ H ˙ s (R d ) with div(u0) = 0 satisfying
NSE has a unique mild solution u ∈ K s,˜ 2,1,T q ∩L ∞ ([0, T]; ˙H s ).
Remark 2.5.8 If in (2.184) we replace the B˙ s−( d
2 − d q ˜ ),∞ ˜ q norm by the H˙ s (R d ) norm, then we get the assumption made in [11, 16] We show that the condition (2.184) is weaker than the condition in [11, 16] In Remark 2.5.6 we showed that
However that these two spaces are dierent Indeed, we have Λ˙ −s | ã | − d 2 ∈/ H˙ s (R d ), on the other hand by using Lemma 1.1.8, we easily prove thatΛ˙ −s | ã | − d 2 ∈B˙ s−( d
Conclusions
In this section, we explore the Sobolev-Lorentz spaces \( H^{\cdot} L_{s,q,r}(\mathbb{R}^d) \) for parameters \( q > 1 \), \( 1 \leq r \leq \infty \), and \( 0 \leq s < d/q \) These spaces are broader than the traditional Sobolev spaces \( H^{\cdot}_{q,s}(\mathbb{R}^d) \), as \( H^{\cdot}_{q,s}(\mathbb{R}^d) = H^{\cdot} L_{s,q,q}(\mathbb{R}^d) \) Additionally, we examine mild solutions to the Navier-Stokes equations (NSE) within the spaces \( L^{\infty}([0, T]; H^{\cdot} L_{s,q,r}(\mathbb{R}^d)) \) when the initial conditions are drawn from the Sobolev-Lorentz spaces.
In the context of the Besov spaces, we establish the existence of mild solutions for the equation H˙ L s q,r(R d ) under the conditions where \(d \geq 2\), \(q > 1\), \(r \geq 1\), \(s \geq 0\), and \(d q - 1 \leq s < d q\) Specifically, for sufficiently small values of \(T\), mild solutions with arbitrary initial values can be obtained, and for any \(T > 0\), mild solutions exist when the initial value's norm is within the Besov spaces.
In the scenario where \( q > d \), \( r = q \), and \( s = 0 \), we derive a result that is more general than the findings of Cannone and Meyer This statement not only surpasses their conclusions but also operates under significantly weaker conditions regarding the initial data.
In the specific scenario where \( q = r = 2 \) and \( d^2 - 1 < s < d^2 \), we derive a result that is more general than those presented by Chemin in [16] and Cannone in [11] This finding offers a stronger statement compared to their work, achieved under significantly weaker conditions on the initial data.
In the case of critical indexes (1< q ≤d, r ≥1, s= d q −1), we get the existence of global mild solutions in spaces L ∞ ([0,∞); ˙H d q −1
L q,r (R d )) when the norm of the initial value in the Besov spacesB˙ d ˜ q −1,∞ ˜ q (R d ), 1 q − 2d 1 < 1 q ˜