Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation

This paper is concerned with the Cauchy problem of 
 three-dimensional modified Navier-Stokes equations with 
 fractional dissipation $ \nu (-\Delta)^{\alpha} u$. 
 The results are three-fold. We first prove the global existence of weak solutions 
 for $0<\alpha\leq1 $ and global smooth solution for 
 $\frac{3}{4}<\alpha\leq1.$ Second, we obtain the optimal decay rates of both weak 
 solutions and the higher-order derivative of the 
smooth solution. 
 Finally, we investigate the asymptotic stability 
of the large solution to the system under large 
initial and external forcing perturbation.


1.
Introduction. Mathematical models for fluid dynamics play an important role in theoretical and computational studies in meteorological, oceanographic sciences and petroleum industries, etc. Naiver-Stokes equations [22] are generally accepted as proving an accurate model for the incompressible motion of viscous fluids in many practical situations, which presume the derivatives of the components of the velocity are small. Since Leray's pioneer work [14] in 1930', however, the question of global regularity or finite-time singularity of three-dimensional Navier-Stokes equations with large initial data is still a big open problem (see [22]). Recently, Caraballo, Real and Kloeden [1] (see also Kloeden, Langa and Real [2,11]) introduced an interesting and important mathematical model which is so-called the globally modification of the Navier-Stokes equations ∂ t u + F N ( ∇u L 2 )(u · ∇u) − ν∆u + ∇p = 0, ∇ · u = 0.
(1. 1) where F N (for some N ∈ R + ) is defined by As stated by Caraballo, Real and Kloeden [1], the system (1.1) is indeed globally modified -the modifying factor F N ( ∇u L 2 ) depends on the norm ∇u L 2 .
Essentially, it prevents large gradients dominating the dynamics and leading to explosions. The system (1.1) maybe violates the basic laws of mechanics, however, on the viewpoint of mathematics, it is a well defined system of equations, just like the modified versions of the Navier-Stokes equations of Leray and others with other mollifications of the nonlinear term (refer to Constantin [3]). Compared with the 3D Naiver-Stokes equations, the system (1.1) in three dimensional case has a unique global smooth solution [1].
Since the presence of the modifying factor F N ( ∇u L 2 ) more or less decreases the singularity of the quadratic convection term u · ∇u, it is possible to control the nonlinear term by using the lower dissipation ν(−∆) α u. Once the observation is right, it is an interesting problem to investigate the new feature such as the wellposedness and large time behavior of solutions compared with the classic Navier-Stokes equations. In this study, we consider this sort globally modified Navier-Stokes equations with fractional dissipation in whole space R 3 ∂ t u + F N ( ∇u L 2 )(u · ∇u) + ν(−∆) α u + ∇p = 0, 0 < α ≤ 1, with the initial condition The main purpose of this study is to give a complete description on the wellposedness and asymptotic behavior of the solutions to system (1.3)-(1.4). More precisely, on one hand, as regards the existence of solutions, when 0 < α ≤ 1, we first construct a global weak solution u ∈ L ∞ (0, T ; L 2 (R 3 )) ∩ L 2 (0, T ; H α (R 3 )) of system (1.3)-(1.4) with u 0 ∈ L 2 (R 3 ) by applying the classic Friedrichs method and Lions-Aubin compactness argument. Second, we prove the local existence of smooth solution u ∈ C([0, T * ); H s (R 3 )) ∩ L 2 (0, T * ; H s+α (R 3 )) and obtain a Beale-Kato-Majda type blow-up criterion under u 0 ∈ H s (R 3 ), s > 3. When 3 4 < α ≤ 1, we prove the global existence of smooth solution to system (1.3)-(1.4). On the other hand, it is desirable to understand the asymptotic behavior of solutions to system (1.3)-(1.4). As respect to the time decay of solutions, by developing the classic Fourier splitting methods introduced by Schonbek [18], we derive the optimal time decay of weak solutions which is more rapid than that of the classic Navier-Stokes equations. In order to investigate the optimal decay estimates of the higher-order derivative of smooth smooth, a new analysis method including iterative technique is employed. Furthermore, since the time decay problem of solutions implies that the trivial solution u = 0 is asymptotic stable, it is an interesting problem to consider the asymptotic stability for the nontrivial solution of system (1.3)-(1.4) with nonzero force. Another objective of this paper is to investigate the asymptotic stability of large solution to the system, we will show the L 2 stability of the difference between the solution of the original system and the perturbed system under large initial data and external forcing perturbation.
To this end, let us introduce the assignment of this paper. In Section 2, we first prove the existence of weak solutions and local smooth solution to system (1.3)-(1.4) with 0 < α ≤ 1, and then further obtain the global smooth solution with 3 4 < α ≤ 1. In Section 3, we prove the optimal L 2 decay rates for both weak solutions and the higher-order derivative of smooth solution. Finally we study the asymptotic stability of the system under large initial and external forcing perturbation in Section 4.

2.
Global smooth solution. In this section we will show that system (1.3)-(1.4) has a global weak solution corresponding to any prescribed L 2 initial data and the global smooth solution for 3 4 < α ≤ 1. To do so, let us first recall some basic facts about Littlewood-Paley decomposition (see [4] for more details). Choose two nonnegative radial functions χ, ϕ ∈ S(R 3 ) supported respectively in B = {ξ ∈ R 3 , |ξ| ≤ 4 3 } and C = {ξ ∈ R 3 , 3 4 ≤ |ξ| ≤ 8 3 } such that for any ξ ∈ R 3 , Let h = F −1 ϕ andh = F −1 χ, the frequency localization operator ∆ j and S j are defined by For any f ∈ S (R 3 ), we have by (2.1) that Before state the main results in this section, we also need some useful lemmas.
Here the constant C ≥ 1 is independent of f and j. (2.7) Now our results read as follows.
Furthermore, if T * is the maximal existence time of the solution, we have the following necessary condition for blow up (2.11) Remark 2.1 It is well-knows that the global smooth solutions of 3D Navier-Stokes equations with large initial data is a big open problem. The main obstacle lies in the fact that the quadratic convection term u · ∇u can not be controlled by the dissipation −ν∆u. Even for 3D Navier-Stokes equations with fractional dissipation Ladyzhenskaya showed the global existence of the smooth solution of (2.12) when α ≥ 5 4 (see also [9]). The main reason why the smooth solution of the system (1.3)-(1.4) here is global only for α > 3 4 is based on our observation that the presence of the modifying factor F N ( ∇u L 2 ) actually decreases the singularity of the quadratic convection term u · ∇u.
Proof of Theorem 2.1 The proof is divided into three steps.
Step 1. Existence of weak solutions We prove the global existence of the weak solution by the classic Friedrichs method which consists of an approximation of (1.3)-(1.4) by a cut-off in the frequency space. Denote J n h = F −1 (χ B(0,n) (ξ)ĥ(ξ)) for n ∈ N and consider the approximate system of (1.3)-(1.4) here P is the projection mapping L 2 onto the subspace {u ∈ L 2 (R 3 ) : ∇ · u = 0}. This is an ODE system on L 2 and the classic Cauchy-Lipschitz theorem ensures that there exists a unique solution which is continuous in time [0, T n ) with value in L 2 . Furthermore, thanks to J 2 n = J n , we claim that J n u n is also a solution of (2.13), so the uniqueness implies that J n u n = u n . Thus u n is also a solution of the following system ∂ t u n + ν(−∆) α u n = −J n P (F N ( ∇u n L 2 )u n · ∇u n ), u n (x, 0) = J n u 0 , (2.14) Noting (1.2), it is easy to verify that the approximate solution u n of (2.14) satisfies 1 2 F N ( ∇u n L 2 )J n (u n · ∇u n ) · u n dx = 0, due to (1.2) and ∇ · u n = 0.
Integrating in time and applying Cauchy inequality gives which ensures that T n = T . Thus there exists a subsequence (denoted by u n again) converges weakly to u such ). But this weak convergence does not allow us to pass to the limit in the nonlinear term J n P (F N ( ∇u n L 2 )u n · ∇u n ). To do so, we need to show ∂ t u n is bounded uniformly in L 4α By using the standard Lions-Aubin compactness theorem [22], which states in our situation that L 2 (0, T ; L 2 (R 3 )) is compactly imbedded in the space Thus the strong convergence of u n ∈ L 2 (0, T ; L 2 (R 3 )) will allow us to show that u is indeed a weak solution of (1.3)-(1.4), which derives the assertion (i) of Theorem 2.1.
Step 2. Local smooth solution and blow-up criterion In order to prove the local existence of the smooth solution, we present the uniform estimate for the approximate solutions u n of (2.14) in H s . Taking the operator ∆ j for j ≥ 0 to both sides of (2.14), we obtain Thanks to ∇ · u n = 0, we have from which, (2.15) and Young's inequality, it follows that and then taking Gronwall inequality into account implies and define T n as From Sobolev embedding inequality, we infer that for 0 ≤ t < T n , Choosing T 1 > 0 such that which contradicts with the definition of T n . Thus there holds for any t ∈ [0, Here we omit the details. Now we prove the Beale-Kate-Majda's blow-up criterion of the smooth solution. Exactly as in the proof of (2.18), we have where we used in the last inequality the fact that ∇u = T(∇ × u) with T a singular integral operator (Biot-Savart law) and Plugging them into (2.21) yields that which together with Gronwall's inequality implies (2.10).
Step 3. Global smooth solution Taking the inner product of (1 Applying Plancherel theorem, Hölder inequality and Young inequality, the right hand side of (2.22) is bounded by Using the product estimates and embedding theorem of the fractional Sobolev spaces gives Thanks to 3 4 < α ≤ 1, i.e. m < m + 1 − α + 1 2 < m + α, applying Gagliardo-Nirenberg inequality and Young inequality yields Plugging (2.25) into (2.24) and applying Gronwall inequality, we have the uniform estimates then there exist two positive constants C 0 and C 1 , such that the solution e −ν(−∆) α t u 0 of the linear equations

2)
with 0 < α ≤ 1 has the following upper and lower bounds Proof of Lemma 3.1 Applying Plancherel theorem and (3.1), it follows that this implies the upper bounds in (3.3), and the lower bounds are derived from Thus we complete the proof of Lemma 3.1.
The following L p − L q estimates are more or less well-known.
is valid for any t > 0.
Proof of Lemma 3.2 Schonbek [21] proved this result for two-dimensional case and the proof in 3D is parallel to the one in 2D. Here we omit the detail.
Our results read as follows. Proof of Theorem 3.1 It should be mentioned that the following discussion should be stated rigorously for the smooth approximated solutions firstly and then taken the limits to get the decay results of the relevant weak solutions. For convenience, we directly discuss for weak solutions. Take the inner product of (1.3) with u, we have and Plancherel theorem gives Denote In order to estimate B(t) | u(ξ, t)| 2 dξ, taking the Fourier transform of (1.3) yields Thus, from which and Plancherel Theorem, (3.8) becomes with m > 0 to be chosen sufficient large and multiplying both sides of above inequalities by (1 + t) m , one shows that d dt (t + 1) m u(t) 2 Integrating (3.10) with respect to t, we get Thanks to (3.6) u(t) L 2 ≤ u 0 L 2 , ∀ t ≥ 0, from which and (3.11), we obtain (3.12) thus we conclude that u(t) L 2 → 0 as t → ∞ which proves the assertion (i).  Remark 3.1 Thanks to the new structure in the nonlinear term with a modifying factor F N ( ∇u L 2 ), here we present a new analysis technique to derive the optimal upper bounds of higher-order derivatives of the smooth solution. One key step here is the assertion (3.18) (see below) and the iteration trick based on (3.18) seems also new. It should be mentioned that the argument here is unavailable for both the classic Navier-Stokes equations and their fractional dissipation form (2.12). On the other hand, the decay rate (3.13) is also more rapid than that of the classic Navier-Stokes equations (see Schonbek [20]).

Upper bounds
Proof of Theorem 3.2 Exactly as in the proof of (2.24), we have (3.14) Thanks to 3 4 < α ≤ 1, i.e. m < m + 1 − α + 1 2 < m + α, letting 0 ≤ k ≤ m and applying Gagliardo-Nirenberg inequality and Young inequality, one shows that Plugging (3.15) into (3.14) becomes Similar to the proof of the case (i) in Theorem 3.1, we have noting Inserting the above inequality into (3.16) and applying (3.5) produce Now, letting g(t) = t+1 β with a suitable large integer β > 0, then multiplying both sides of (3.17) by (1 + t) β and integrating with respect to t, it follows that Employing (3.5) to the right hand of (3.18) with k = 0, we get Consider the integral equation of (1.3)

20)
Taking ∇ k to both sides of equation (3.20) and applying Lemma 3.1, one shows that For I, using Lemma 3.2 and Theorem 3.1, we have 22) and for II Plugging (3.21)-(3.22) into the right hand side of (3.21), we derive In particular, we derive an improved decay rate with respect to (3.19) Repeating this process N times till then we reach the desired decay results Hence the proof of Theorem 3.2 is complete.
u(x, 0) = u 0 (4.1) and the perturbed system with the initial data and external forcing perturbation w 0 (x) and g(x, t), we are focused on the asymptotic stability of system (4.1) with external forcing f (x, t) under the large initial data and external forcing perturbation w 0 and g. Our result reads as follows.
is a global solution of the original system (4.1), then the asymptotic stability property holds true for every weak solution v(x, t) of the perturbed equations (4.2).

Remark 4.1
The proofs of the global existence of the solution for both the original system (4.1) and the perturbed system (4.2) in Theorem 4.1 are parallel to the ones in Theorem 2.1 in Section 2. It should be mentioned that since nonzero force f (x, t) ∈ L 2 (0, T ; L 2 (R 3 )), the original system (4.1) has a nontrivial solution u = 0 which has not any decay property according to Remark 3.2 in Section 3. The same assertion is also valid for the perturbed system (4.2).

Remark 4.2
Compared with the asymptotic stability results of the 3D Navier-Stokes equations by Kawanago [10], Ponce et al [17], Kozono [13], Zhou [23], neither additional assumption on the solutions nor small assumption on initial and external forcing perturbation is added in Theorem 4.1. The key in the proof of Theorem 4.1 is to prove the global estimate of the difference u(t) − v(t) and an auxiliary decay estimate by choosing a suitable test function which was first introduced by Masuda [16] (see also Dong and Chen [5]).
Proof of Theorem 4.1 The proof is divided into three steps.
Step 1 An auxiliary identity This section is devoted to the derivation of the following auxiliary identity the proof of (4.4) is based on a special choice of test functions. The idea of choosing test functions is developed from the argument of Masuda [16]. We introduce the test functions for the solutions u and v. We define the mollifier function η ε (ε > 0) expressed as It follows from the definition of the weak solutions with the choosing test functions v ε and u ε that Combination of these two equations (4.6) and (4.7) yields where we have used the following observation due to the Fubini theorem and the symmetry of η ε . We now prove the convergence of the equation (4.8) to the desired identity (4.4) as ε → 0. Firstly, for the right-hand side of (4.8), due to the properties of the mollifier function η ε , therefore, it is readily deduced that With the aid of the weak continuity of u and v in L 2 (R 3 ) and the definition of the function η ε , one shows that Similarly, we have For the left-hand side of (4.8), since due to the properties of the mollifier function η ε , therefore, it is readily deduced that Now it remains to prove the validity of the following convergence results By using Hölder inequality, Gagliardo-Nirenberg inequality, we have Hence, letting ε → 0 in (4.8), we have the auxiliary identity.
Step 2 Global estimates of u(x, t) − v(x, t) Denote by w(x, t) = u(x, t) − v(x, t) the difference between the solution u(x, t) of the original system (4.1) and the solution v(x, t) of the perturbed equations (4.2).
Multiplying (4.1) by u(x, t) and integrate over R 3 , we obtain and, likewise, for global weak solution v(x, t), the summation of (4.9-4.10) gives that On the other hand, the difference w(x, t) = u(x, t) − v(x, t) also satisfies the energytype inequality Therefore the substitution of (4.11) into (4.12) produces With the use of the auxiliary identity (4.4), one shows that (4.14) By using Hölder inequality and Gagliardo-Nirenberg inequality, one shows that for the right hand side of (4.14) one by one and Inserting (4.15-4.17) into (4.14) to deduce (4.18) Taking Gronwall inequality into consideration gives for the constant C independent of t > 0, where we have used the bounds of the solution u with initial data u 0 ∈ H 1 Furthermore, the derivation of (4.18) and (4.20) also implies the estimate, for 0 ≤ s < t < ∞ Step 3 Proof of (4.3) In order to prove the asymptotic stability we need the following auxiliary decay estimate We now estimate (4.24) one by one. Firstly, for the first two terms of the left hand side of (4.24), by the definition of η (w(t), ϕ (t)) − (w(s), ϕ (s)) = t s η (t − σ)(U (t)w(t), U (σ)w(σ))dσ − t s η (s − σ)(U (s)w(s), U (σ)w(σ))dσ (4.28) Secondly, for others of the left hand side of (4.24), we obtain that  Thus, inserting (4.28-4.32) into (4.24) and letting → 0, we obtain which implies the desired auxiliary decay estimate lim sup after letting t → ∞ and then s → ∞.
Using Gagliardo-Nirenberg inequality yields as t → ∞. Hence we complete the proof of Lemma 4.1.