GLOBAL REGULARITY OF THE TWO-DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM WITH ZERO ANGULAR VISCOSITY
|
|
- Edward Hardy
- 6 years ago
- Views:
Transcription
1 GLOBAL REGULARITY OF THE TWO-DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM WITH ZERO ANGULAR VISCOSITY KAZUO YAMAZAKI Abstract. We study the two-dimensional magneto-micropolar fluid system. Making use of the structure of the system, we show that with zero angular viscosity the solution triple remains smooth for all time. Keywords: Magneto-micropolar fluid, micropolar fluid, global regularity, regularity criteria, Besov spaces. Introduction The magneto-micropolar fluid (MMPF) system consists of the following euations: u t + (u )u r(b )b + (π + r b ) + (µ + χ)λ r u = χ( w), (a) ρ w t + ρ(u )w + w = χw + (α + β) divw + χ( u), γλr (b) b t + (u )b (b )u + νλr3 b =, (c) u = b =, (u, w, b)(x, ) = (u, w, b )(x), (d) where we denoted u, w, b, π the velocity, micro-rotational velocity, the magnetic and the hydrostatic pressure fields respectively. We also denoted physically meaningful uantities: r = M ReRm where M is the Hartmann number, Re the Reynolds number, Rm the magnetic Reynolds number, χ the vortex viscosity, µ the kinematic viscosity, ρ the microinertia, α, β, γ the angular viscosities, ν = Rm all of which we assume to be positive taking into account of conditions such as Clausius-Duhem ineuality. Finally, we denoted fractional Laplacians defined through Fourier transform by Λ ri f(ξ) = ξ ri ˆf(ξ), i =,, 3. Hereafter let us write t = t, i = x i and also assume ρ =. Let us discuss the rich history of mathematical and physical study concerning the MMPF system and its related systems. Firstly, the MMPF system at w b The author expresses gratitude to Professor Jiahong Wu and Professor David Ullrich for their teaching. MSC : 35B65, 35Q35, 35Q86 Department of Mathematics, Oklahoma State University, 4 Mathematical Sciences, Stillwater, OK 7478, USA
2 KAZUO YAMAZAKI reduces to the Navier-Stokes euations (NSE). The NSE has ample engineering applications in fluid mechanics and has been investigated mathematically with much intensity. Secondly, the MMPF system at χ =, w reduces to the magnetohydrodynamics (MHD) system which describes the motion of electrically conducting fluids and has broad applications in applied sciences such as astrophysics, geophysics and plasma physics (cf. [3]). The mathematical analysis of the MHD system has attracted much attention in particular concerning the global regularity issue in two-dimensional case (cf. [3, 4, 7, 4, 7, 3, 3, 33, 38]). Thirdly, the MMPF system at b reduces to the micropolar fluid (MPF) system: t u + (u )u + π + (µ + χ)λ r u = χ( w), t w + (u )w + γλ r w = χw + (α + β) divw + χ( u), (a) (b) u =, (u, w)(x, ) = (u, w )(x). (c) The microfluids and micropolar fluids were introduced in [3, 4] respectively. In particular, the micropolar fluids represent the fluids consisting of bar-like elements, e.g. anisotropic fluids, such as liuid crystals made up of dumbbell molecules and animal blood. The study of this system was continued by many (e.g. [5, 6, 8, 9,,, 8, 9, 9, 3, 37]). In particular, the authors in [] obtained global regularity result of the two-dimensional MPF system with zero angular viscosity and r =. Finally, the MMPF system (a)-(d) was considered in [] in which the authors obtained Serrin-type stability criteria. This system has also found much applications and attraction by mathematicians (cf. [6,,, 5, 8, 3, 35, 36, 39]). In particular, the authors in [5] obtained the global existence of weak solution triple (u, w, b), global existence of strong solution triple (u, w, b) in case initial data is small if dimension is three, while the uniue weak solution triple (u, w, b) if dimension is two, both with r = r = r 3 =. We now consider, following the work of [, ], the two-dimensional problem by setting and denote by u = (u, u, ), w = (,, w 3 ) Ω := u = u u, j =: b = b b, w = ( w, w). We present our results concerning the following special case of () with x R : t u + (u )u (b )b + (π + b ) = (µ + χ) u + χ( w), t w + (u )w = χw + χ( u), t b + (u )b (b )u = ν b. (3a) (3b) Theorem.. For every (u, w, b ) H s (R ), s >, there exists a uniue solution triple (u, w, b) to (3a)-(3c) such that u, b C([, ); H s (R )) L ([, ); H s+ (R )), w C([, ); H s (R )). (3c)
3 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM 3 Remark.. () Theorem. improves previous global regularity results with angular viscosity (cf. []) and extend those in [] from MPF system to the MMPF system. Furthermore, it is also an extension of the global wellposedness result of the two-dimensional classical MHD system (cf. [3]). () In three-dimension, employing standard energy method on (a)-(d), one can show that the global regularity is attained as long as r + N, min{r + r, r + r 3 } + N, N = 3, 4 (cf. [6]). Hence, the result of Theorem. at r = r 3 =, r = is a significant improvement, attainable due to the advantage of the structure of the system upon taking curls, very similarly to the recent developments in the two-dimensional generalized MHD system. (3) Local existence proof can be performed in many ways, for example using mollifiers following the work on the NSE. Because the authors in [] proved the local existence in the case of the MPF system in detail and one can follow their work line by line, just adding the magnetic field euation, we omit this proof. Let us use the notation A a,b B, A a,b B to imply that there exists a nonnegative constant c that depends on a, b such that A cb, A = cb and briefly discuss the difficulty of this problem before we present our proof. Firstly, we take L -inner products with (u, w, b) on (3a)-(3c) respectively to obtain due to the incompressibility of u, t u L + (µ + χ) u L = (b )b u + χ ( w) u t w L + χ w L = χ ( u)w t b L + ν b L = (b )u b In sum, using the fact that ( w) u = ( u)w, we obtain t( u L + w L + b L ) + (µ + χ) u L + χ w L + ν b L χ w L u L χ w L + χ u L by the incompressibility of b, Hölder s and Young s ineualities. Therefore, after absorbing the right hand side, integrating in time we obtain sup t [,T ] ( u L + w L + ) T b L (t) + u L + b L dτ u,w,b. (4) We will obtain via a commutator estimate a regularity criteria that in order to prove Theorem., it suffices to show the time integrability of Ω L, w L, j L (See Proposition 3.). Now taking a curl on (3a) leads to t Ω (µ + χ) Ω = (u )Ω + (b )j χ w. (5)
4 4 KAZUO YAMAZAKI Due to the lack of angular viscosity, we have no obvious way to handle χ w. The key observation in [] was that by defining ( ) χ Z := Ω w, (6) µ + χ we can take advantage of its evolution in time governed by t Z = (µ + χ) Z (u )Z c Z + c w + (b )j (7) where we denoted c := χ µ + χ, c := χ µ + χ χ 3 (µ + χ). This leads to the following L p -estimate of Z after multiplying (7) by Z p Z, integrating in space, using the incompressibility of u and Hölder s ineuality: p t Z p L w p L p Z p L + (b )j p L p Z p L. p Dividing by Z p Lp, we obtain taking limit p, t Z L w L + (b )j L w L + b L j L. (8) On the other hand, from (3b) we can estimate similarly p t w p L + p χ w p L χ Ω p L p w p L Z p L p w p L + p w p L p. by the incompressibility of u, Hölder s ineuality and (6). Dividing by w p L p taking limit p lead to t w L Z L + w L. (9) In sum of (8) and (9) we obtain t ( Z L + w L ) Z L + w L + b L j L. Thus, in the absence of (b )j, the authors in [] were able to show the global regularity for the two-dimensional MPF system with zero angular viscosity. If we replace the diffusive term b by Λ r3 b so that this becomes familiar to the recent developments in the two-dimensional generalized MHD system (cf. [3]), then this problem in our case reuires, according to (4), r 3 > 3 considering a Sobolev embedding H r3 (R ) L (R ). A natural idea to reduce the lower bound on r 3 is to improve the bound on b. To create some cancellations, it is optimal to estimate Ω and j together. Because taking a curl on (3c) gives t j ν j = (u )j + (b )Ω + [ b ( u + u ) u ( b + b )], () we have the L -estimates of Ω and j as follows: t Ω L + (µ + χ) Ω L = (b )jω χ wω, t j L + ν j L = (b )Ωj + [ b ( u + u ) u ( b + b )]j. and
5 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM 5 Considering the term χ wω, we have no choice but to try the estimate on w simultaneously. However, taking L -inner products of (3b) with w leads to t w L + χ w L = u w w + χ Ω w, after integration by parts and the first nonlinear term becomes problematic. The novelty of this manuscript is the observation that by defining Z in (6), a certain bound of j will lead to a bound of Z, which in turn leads to the bound of w, and then Ω. In the next section, we set up notations and state key lemmas. Thereafter we prove our result. The proof consists of two parts: a regularity criteria for the solution triple to (3a)-(3c) using a commutator estimate and an a priori estimate making use of Z in (6).. Preliminaries We state some key lemmas and facts that will be crucial in our proofs. Lemma.. (cf. [7]) Let f be divergence-free vector field such that f L p, p (, ). Then f L p c p p curl f L p. The next lemma consists of variations of the classical result from []; for the proofs of these specific ineualities we refer readers to the Appendices of [3, 34]: Lemma.. Suppose f L (R ) H s (R ), s >. Then f L c ( f L + curl f L log ( + f H s) + ), () f L c ( f L + f H log ( + f H s) + ). () Let us recall the notion of Besov spaces (cf. [7]). We denote by S(R ) the Schwartz class functions and S (R ), its dual. We define S to be the subspace of S in the following sense: S = {φ S, φ(x)x l dx =, l =,,,...}. R Its dual S is given by S = S/S = S /P where P is the space of polynomials. For k Z we define A k = {ξ R : k < ξ < k+ }. It is well-known that there exists a seuence {Φ k } S(R ) such that supp ˆΦ k A k, ˆΦk (ξ) = ˆΦ ( k ξ) or Φ k (x) = k Φ ( k x) and k= Conseuently, for any f S, ˆΦ k (ξ) = { if ξ R \ {}, if ξ =.
6 6 KAZUO YAMAZAKI k= Φ k f = f. To define the homogeneous Besov spaces, we set k f = Φ k f, k =, ±, ±,... With such we can define for s R, p, [, ], the homogeneous Besov spaces where f Ḃs p, = Ḃp, s = {f S : f Ḃs < }, p, { ( k (ks k f L p) ) if <, sup k ks k f L p if =. To define the inhomogeneous Besov spaces, we let Ψ C (R ) be such that = ˆΨ(ξ) + ˆΦ k (ξ), Ψ f + Φ k f = f, k= for any f S. With that, we set k= if k, k f = Ψ f if k =, Φ k f if k =,,,..., and define for any s R, p, [, ], the inhomogeneous Besov spaces where f B s p, = For any s R, < p <, we have Bp, s = {f S : f B s p, < }, { ( k= (ks k f L p) ), if <, sup k< ks k f L p if =. B s p,min{p,} W s,p B s p,max{p,}, Ḃs p,min{p,} Ẇ s,p Ḃs p,max{p,}. Moreover, Bony s paraproduct decomposition will be used freuently (cf. [7]) fg = T f g + T g f + R(f, g) where T f g = k S k f k g, R(f, g) = k f k g, S k = m. k,k : k k m:m k 3. Commutator estimate and regularity criteria In this section, we obtain a regularity criteria for the solution (u, w, b) to (3a)-(3c) using a commutator estimate, namely the following:
7 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM 7 Proposition 3.. Let s >, f H s (R ) L (R ), g H s (R ) L (R ), g =. Then for any k 3, [ k, g i i ]f L c k ks ( g L f H s + g H s f L ). (3) Moreover, for s > if additionally f L (R ), then for any k, [ k, g i i ]f L c k ks ( g L f H s + g H s f L ) (4) where {c k } l k. Proof. The ineuality (3) is shown in []; we focus on (4). We first write [ k, g i i ]f = k (T g i i f + T if g i + R(g i, i f)) (T g i i k f + T i k f g i + R(g i, i k f)) =[ k, T g i i ]f + k (T if g i ) + k (R(g i, i f)) T i k f g i R(g i, i k f) where we used the Bony s paraproduct decomposition. Furthermore, we write [ k, T g i i ]f = k We rewrite this as k = = = = k : k k k : k k k : k k k : k k k : k k k : k k S k g i k i f S k g i k i f k : k k k : k k (S k g i ) k ( k i f) [(S k g i ) k ( k i f) k (S k g i k i f)] R k Φ ( k y)[ ( ) k Φ ( k y) y i R 3k R (5) (S k g i ) k ( k i f). τ S k g i (x τy)dτ] k f(x y)dy y i [ τ S k g i (x τy)dτ] k f(x y)dy y S k g i (x τy)dτ Φ ( k y) ( k y) i k f(x y)dy. by definition of k and integration by parts using divergence-free property of g. Thus, [ k, T g i i ]f L k : k k S k g i L 3k x Φ ( k x) i ( k x) by Young s ineuality for convolution. Hence, because Φ S(R ), L k f L [ k, T g i i ]f L g L k f L c k ks g L f H s (6) where we used that under restriction of k k, we may replace k by k modifying constants. Next,
8 8 KAZUO YAMAZAKI k (T if g i ) L S k f L k g L k : k k (7) f L k g L c k ks f L g H s by Hölder s ineuality. Next, k R(g i, i f) L k g i k i f L f L ks k :k k +3 k,k k +3 (k k )s k s k g L c k ks f L g H s (8) by Hölder s ineuality. Next, T i k f g i L = ( k ( i k f)) k g i L k k :k k k :k k k i f k g i L c k ks f L g H s (9) where we used that k k f = if k k > and Hölder s ineuality. Finally, R(g i, i k f) L k g i k i k f L k,k : k k () f L k g L c k ks f L g H s by Hölder s ineualities. obtain Taking L -norm on (5) and considering (6)-(), we [ k, g i i ]f L c k ks ( f L g H s + g L f H s). This completes the proof of Proposition 3.. Proposition 3.. Suppose (u, w, b ) H s (R ), s > and its corresponding solution triple (u, w, b) to (3a)-(3c) in time interval [, T ] satisfies T Ω L + w L + j Ldτ where Ω = u, j = b. Then (u, w, b) H s (R ) for all time t [, T ]. Proof. We apply k, k and take L -inner products of (3a)-(3c) with ( k u, k w, k b) respectively and sum to obtain
9 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM 9 t( k u L + kw L + kb L ) + (µ + χ) k u L + χ kw L + ν kb L = k ((u )u) k u (u ) k u k u + k ((b )b) k u (b ) k b k u k ((u )w) k w (u ) k w k w k ((u )b) k b (u ) k b k + k ((b )u) k b (b ) k u k b + χ k w k ( u) () where we used the incompressibility of u and b. For k 3, we now bound t( k u L + kw L + kb L ) + (µ + χ) k u L + χ kw L + ν kb L = [ k, u ]u k u + [ k, b ]b k u [ k, u ]w k w [ k, u ]b k b + [ k, b ]u k b + χ k w k ( u) [ k, u ]u L k u L + [ k, b ]b L k u L + [ k, u ]w L k w L + [ k, u ]b L k b L + [ k, b ]u L k b L + χ k w L k u L c k ks (( u L u H s + b L b H s) k u L + ( u L w H s + u H s w L ) k w L + ( u L b H s + b L u H s) k b L ) + χ k w L + χ ku L by Hölder s ineualities, (4) on the first, second, fourth and fifth commutators and (3) on the third and Young s ineualities. Absorbing the last two terms, multiplying by ks, k 3, s >, we obtain t ks ( k u L + kw L + kb L ) + (µ + χ) ks k u L + νks k b L c k ks ( k u L + k w L + k b L ) ( u H s + w H s + b H s + )( u L + b L + ) + c k ks u H s w L k w L.
10 KAZUO YAMAZAKI For the case k =,,,, we use the fact that k is a convolution operator, Ψ, Φ k S(R ), Hölder s and Young s ineualities for convolution and (4) to obtain k= k= ks k ((u )u) k u + k ((u )b) k b + k ((b )b) k u k ((b )u) k b + χ ks ( u u L u L + b b L u L + u w L w L + u b L b L + b u L b L + w L u L ). k ((u )w) k w k w k ( u) Thus, we can multiply () by ks, sum over k and apply Hölder s ineualities to obtain t ( u H s + w H s + b H s) + (µ + χ) u H s + ν b H s ( + c k l )( u H s + w H s + b H s + )( u L + b L + ) + ( + c k l ) u H s w L w H s. () For the last term on the right hand side, we take Young s ineualities to obtain c( + c k l ) u H s w L w H s µ + χ u H + s c w L w Hs. (3) We apply (3) to (), absorb the first term on the right hand side of (3) to obtain t ( u H s + w H s + b H s) ( u H s + w H s + b H s + )( u L + w L + b L + ). By () of Lemma. applied to u L, () of Lemma. applied to b L and (4) we obtain t log ( + u H s + w H s + b H s) log ( + u H s + w H s + b H s)( + Ω L + w L + j L ). Gronwall s ineuality completes the proof of Proposition A priori estimate We consider Z from (6) and first obtain L -estimates. Proposition 4.. Suppose (u, w, b) solves (3a)-(3c) in time interval [, T ]. Then sup t [,T ] ( Z L + Ω L + j L )(t) + T Z L + j Ldτ.
11 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM Proof. We take L -inner products (7) and () with (Z, j) respectively to obtain t Z L +(µ + χ) Z L + c Z L c w L Z L + (b )j ( ( ) ) χ Ω w, µ + χ t j L +ν j L = (b )Ωj + [ b ( u + u ) u ( b + b )]j. (4) (5) where we used (6). We sum (4) and (5) and bound t( Z L + j L ) + (µ + χ) Z L + ν j L + c Z L Z L + b L j L w L + b L 4 u L j L 4 Z L + b L j 3 L + j L j L u L ν j L + c( Z L + + j L Ω L ) ν j L + c( Z L + + j L ( Z L + w L )) ν j L + c( Z L + j L )( + j L ) by Hölder s and Gagliardo-Nirenberg ineualities, (4), Lemma., Young s ineualities and (6). By Gronwall s ineuality and (4), this implies sup t [,T ] ( Z L + j L ) (t) + T Z L + j Ldτ. Because we have the bound on w(t) L from (4), by (6) we obtain the bound on Ω(t) L. Thus, the proof of Proposition 4. is complete. We obtain slightly higher integrability now: Proposition 4.. Suppose (u, w, b) solves (3a)-(3c) in time interval [, T ]. Then for any 4, sup ( Z L + Ω L + j L )(t). (6) t [,T ] Proof. We assume < 4 below as the case = is already done in the Proposition 4.. We multiply (7) by Z Z, integrate in space to obtain t Z L (µ + χ) Z Z Z + c Z L =c w Z Z + (b )j Z Z. For the dissipative term, we integrate by parts to obtain (7)
12 KAZUO YAMAZAKI (µ + χ) On the other hand, Z Z Z = 4(µ + χ)( ) Z L. (8) Z L ( ) = Z L 4( ) Z ( L 4 ) Z ( ) L Z ( L ( ) ) where we used the Gagliardo-Nirenberg ineuality and Proposition 4.. This implies Now Z L ( ) Z ( ) L. (9) c w Z Z + (b )j Z Z c w L Z ( ) b ( Z) Z j L ( ) c Z ( ) ( ) ( ) L b Z Z j ( ) (µ + χ)( ) Z L + c + b Z j by Hölder s ineuality, (4), (9), integration by parts and Young s ineuality. We can absorb the dissipative term whereas for the second term we estimate b Z j b L Z L j L b L j L Z L j ( ) L ( j ) L (3) +( j L ( Z L + ) (3) by Hölder s and Gagliardo-Nirenberg ineualities, (4) and Proposition 4.. Thus, we have after absorbing, from (7), (8), (3) and (3), t Z (µ + χ)( ) L + Z L ( j L + )( Z L + ) (3) by Young s ineuality. Thus, by Proposition 4. and Gronwall s ineuality applied to (3) we obtain sup Z(t) L. (33) t [,T ] Next, we multiply (3b) by w w, integrate in space to obtain t w L + χ w L Ω L w L Z L w L + w L ( + w L ) by Hölder s ineualities, (6) and (33). This implies by Gronwall s ineuality and hence by (6) and (33), sup w(t) L t [,T ]
13 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM 3 sup Ω(t) L. (34) t [,T ] Finally, we multiply () by j j and integrate in space to estimate t j L ν j j j = (b )Ω j j + [ b ( u + u ) u ( b + b )] j j. On the diffusive term, we apply integration by parts, ν j j j = ν( ) j j j = ν( ) j j (35) (36) and estimate the first non-linear term of (35) by integration by parts and Young s ineuality as follows: (b )Ω j j = ( ) ν( ) 4 b j j Ω + c b j Ω. j j We can absorb the diffusive term whereas for the second term of (37), c b j Ω b L j L Ω L (37) b L j L j L ( + j L )( j L + ) (38) by Hölder s and Gagliardo-Nirenberg ineualities, (34) and (4). Moreover, we can also write On the other hand, ν( ) j j ( j L = j L j 4 = L 4 ν( )4 j L. (39) ) j L j ( ) L by the Gagliardo-Nirenberg ineualities and Proposition 4. and therefore j L j 4( ) L. (4) Now we estimate the second integral in (35): [ b ( u + u ) u ( b + b )] j j b L u L j L j L j L j L Ω L j 4( ) L j ν( ) L j L + c( j L + ) (4)
14 4 KAZUO YAMAZAKI by Hölder s ineualities, Lemma., (34), (4) and Young s ineualities. Therefore, from (35)-(39), (4), after absorbing we obtain t j ν( ) j j L + ν( ) + 4 j L (+ j L )( j L +). By Proposition 4., Gronwall s ineuality completes the proof of Proposition 4.. We now show that the criteria according to Proposition 3. is satisfied Bound of w(t) L. Firstly, we take L -inner products on (7) with Z to obtain t Z L + (µ + χ) Z L + c Z L = (u )Z Z c w Z (b )j Z u L Z L Z L + w L Z L + b L j L Z L u 4 L u 3 4 L 3 Z L Z L + Z L + b 4 L b 3 4 L 3 j L Z L µ + χ Z L + c( + Z L )( + j L ) by Hölder s ineualities, (4), the Gagliardo-Nirenberg and Young s ineualities and Proposition 4.. After absorbing, by Gronwall s ineuality we obtain sup t [,T ] Z(t) L + T Z Ldτ (4) due to Proposition 4.. Next, we multiply (3b) by w p w, integrate in space to obtain p t w p L + p χ w p L Ω p L p w p L Z p L p w p L + p w p L p by Hölder s ineualities and (6). Dividing by w p, we obtain w(t) L p w L pe ct + Taking limit p, we obtain t L p t e c(t τ) Z L pdτ t + e T Z L pdτ. T w(t) L T + e T Z L dτ. Therefore, by Proposition 4., (4) and Sobolev embedding, we obtain sup w(t) L. (43) t [,T ] 4... Bound of Ω(t) L. By (4), (43) and (6), we immediately obtain T Ω(t) L dτ T Z(t) L dt + T w(t) L dt. (44)
15 TWO DIMENSIONAL MAGNETO-MICROPOLAR FLUID SYSTEM Bound of j(t) L. By Hölder s ineuality and Proposition 4., T j L dτ T ( T This completes the proof of Theorem.. References j L dτ ). [] G. Ahmadi, and M. Shahinpoor, Universal stability of magneto-micropolar fluid motions, Int. J. Engng. Sci.,, (974) [] H. Brezis, and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution ineualities, Comm. Partial Differential Euations, 5, 7 (98), [3] C. Cao, and J. Wu, Global regularity for the D MHD euations with mixed partial dissipation and magnetic diffusion, Adv. Math., 6 (), [4] C. Cao, J. Wu, and B. Yuan, The D incompressible magnetohydrodynamics euations with only magnetic diffusion, SIAM J. Math. Anal.,, 46 (4), [5] M. Chen, Global well-posedness of the D incompressible micropolar fluid flows with partial viscosity and angular viscosity, Acta Math. Sci. Ser. B Engl. Ed., 33B, 4 (3), [6] Q. Chen, and C. Miao, Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differential Euations, 5 (), [7] J.-Y. Chemin, Perfect incompressible fluids, Oxford lecture series in mathematics and its applications, 4 (998). [8] B.-Q. Dong, and Z.-M. Chen, Regularity criteria of weak solutions to the three-dimensional micropolar flows, J. Math. Phys., 5, 355 (9). [9] B.-Q. Dong, and Z.-M. Chen, Asymptotic profiles of solutions to the D viscous incompressible micropolar fluid flows, Discrete Contin. Dyn. Syst., 3, 3 (9), [] B.Q. Dong, Y. Jia, and Z.-M. Chen, Pressure regularity criteria of the three-dimensional micropolar fluid flows, Math. Methods Appl. Sci., 34 (), [] B.-Q. Dong, and W. Zhang, On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces, Nonlinear Anal., 73 (), [] B.-Q. Dong, and Z. Zhang, Global regularity of the D micropolar fluid flows with zero angular viscosity, J. Differential Euations, 49 (), -3. [3] A. C. Eringen, Simple microfluids, Int. Engng. Sci.,, (964), 5-7. [4] A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 6 (966), -8. [5] G. P. Galdi, and S. Rionero, A note on the existence and uniueness of solutions of the micropolar fluid euations, Int. J. Engng. Sci., 5 (977), 5-8. [6] H. Inoue, K. Matsuura, and M. Ǒtani, Strong solutions of magneto-micropolar fluid euation, Discrete and continuous dynamical systems, 3; Dynamical systems and differential euations (Wilmington, NC, ), [7] Q. Jiu, and J. Zhao, Global regularity of D generalized MHD euations with magnetic diffusion, arxiv: [math.ap], 8, Nov., 3. [8] G. Lukaszewicz, On nonstationary flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., XII, 6 (988), [9] G. Lukaszewicz, On the existence, uniueness and asymptotic properties for solutions of flows of asymmetric fluids, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., XIII, 6 (989), 5-. [] G. Lukaszewicz, Micropolar fluids: theory and applications, Birkhäuser, 999. [] E. E. Ortega-Torres, and M. A. Rojas-Medar, Magneto-micropolar fluid motion: global existence of strong solutions, Abstr. Appl. Anal., 4, (999), 9-5. [] M. A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniueness of strong solutions, Math. Nachr., 88 (997), [3] M. Sermange, and R. Temam, Some mathematical uestions related to the MHD euations, Comm. Pure Appl. Math., 36 (983), [4] C. V. Tran, X. Yu, and Z. Zhai, On global regularity of D generalized magnetohydrodynamics euations, J. Differential Euations, 54, (3), [5] Y. Wang, Regularity criterion for a weak solution to the three-dimensional magneto-micropolar fluid euations, Bound. Value Probl., 58 (3). [6] J. Wu, The generalized MHD euations, J. Differential Euations, 95 (3), 84-3.
16 6 KAZUO YAMAZAKI [7] J. Wu, Global regularity for a class of generalized magnetohydrodynamic euations, J. Math. Fluid Mech., 3, (), [8] Z. Xiang, and H. Yang, On the regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative, Bound. Value Probl., 39 (). [9] L. Xue, Wellposedness and zero microrotation viscosity limit of the D micropolar fluid euations, Math. Methods Appl. Sci., 34 (), [3] N. Yamaguchi, Existence of global strong solution to the micropolar fluid system in a bounded domain, Math. Meth. Appl. Sci., 8 (5), [3] K. Yamazaki, Remarks on the global regularity of two-dimensional magnetohydrodynamics system with zero dissipation, Nonlinear Anal., 94 (4), [3] K. Yamazaki, Global regularity of logarithmically supercritical MHD system with zero diffusivity, Appl. Math. Lett., 9 (4), [33] K. Yamazaki, On the global regularity of two-dimensional generalized magnetohydrodynamics system, J. Math. Anal. Appl., 46 (4), 99-. [34] K. Yamazaki, On the global regularity of N-dimensional generalized Boussines system, Appl. Math., to appear. [35] K. Yamazaki, (N ) velocity components condition for the generalized MHD system in N- dimension, Kinet. Relat. Models, to appear. [36] B. Yuan, Regularity of weak solutions to magneto-micropolar fluid euations, Acta Math. Sci. Ser. B Engl. Ed., 3B, 5 (), [37] B. Yuan, On regularity criteria for weak solutions to the micropolar fluid euations in Lorentz space, Proc. Amer. Math. Soc., 38, 6 (), [38] B. Yuan, and L. Bai, Remarks on global regularity of D generalized MHD euations, J. Math. Anal. Appl., 43, (4), [39] J. Yuan, Existence theorem and blow-up criterion of the strong solutions to the magnetomicropolar fluid euations, Math. Meth. Appl. Sci., 3 (8), 3-3.
GLOBAL REGULARITY OF LOGARITHMICALLY SUPERCRITICAL 3-D LAMHD-ALPHA SYSTEM WITH ZERO DIFFUSION
GLOBAL REGULARITY OF LOGARITHMICALLY SUPERCRITICAL 3-D LAMHD-ALPHA SYSTEM WITH ZERO DIFFUSION KAZUO YAMAZAKI Abstract. We study the three-dimensional Lagrangian-averaged magnetohydrodynamicsalpha system
More informationSECOND PROOF OF THE GLOBAL REGULARITY OF THE TWO-DIMENSIONAL MHD SYSTEM WITH FULL DIFFUSION AND ARBITRARY WEAK DISSIPATION
SECOND PROOF OF THE GOBA REGUARITY OF THE TWO-DIMENSIONA MHD SYSTEM WITH FU DIFFUSION AND ARBITRARY WEAK DISSIPATION KAZUO YAMAZAKI Abstract. In regards to the mathematical issue of whether a system of
More informationRecent developments on the micropolar and magneto-micropolar fluid systems: deterministic and stochastic perspectives
Recent developments on the micropolar and magneto-micropolar fluid systems: deterministic and stochastic perspectives Kazuo Yamazaki Abstract We review recent developments on the micropolar and magneto-micropolar
More informationON THE GLOBAL REGULARITY ISSUE OF THE TWO-DIMENSIONAL MAGNETOHYDRODYNAMICS SYSTEM WITH MAGNETIC DIFFUSION WEAKER THAN A LAPLACIAN
ON THE GOBA REGUARITY ISSUE OF THE TWO-DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEM WITH MAGNETIC DIFFUSION WEAKER THAN A APACIAN KAZUO YAMAZAKI Abstract. In this manuscript, we discuss the recent developments
More informationNonlinear Analysis. A regularity criterion for the 3D magneto-micropolar fluid equations in Triebel Lizorkin spaces
Nonlinear Analysis 74 (11) 5 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A regularity criterion for the 3D magneto-micropolar fluid equations
More informationThe 2D Magnetohydrodynamic Equations with Partial Dissipation. Oklahoma State University
The 2D Magnetohydrodynamic Equations with Partial Dissipation Jiahong Wu Oklahoma State University IPAM Workshop Mathematical Analysis of Turbulence IPAM, UCLA, September 29-October 3, 2014 1 / 112 Outline
More informationc 2014 Society for Industrial and Applied Mathematics
SIAM J. MATH. ANAL. Vol. 46, No. 1, pp. 588 62 c 214 Society for Industrial and Applied Mathematics THE 2D INCOMPRESSIBLE MAGNETOHYDRODYNAMICS EQUATIONS WITH ONLY MAGNETIC DIFFUSION CHONGSHENG CAO, JIAHONG
More informationON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS
ON THE GLOBAL REGULARITY OF GENERALIZED LERAY-ALPHA TYPE MODELS KAZUO YAMAZAKI 1 2 Abstract. We generalize Leray-alpha type models studied in [3] and [8] via fractional Laplacians and employ Besov space
More informationREGULARITY CRITERIA FOR WEAK SOLUTIONS TO 3D INCOMPRESSIBLE MHD EQUATIONS WITH HALL TERM
Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 10, pp. 1 12. ISSN: 1072-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EGULAITY CITEIA FO WEAK SOLUTIONS TO D INCOMPESSIBLE
More informationarxiv: v2 [math.ap] 30 Jan 2015
LOCAL WELL-POSEDNESS FOR THE HALL-MHD EQUATIONS WITH FRACTIONAL MAGNETIC DIFFUSION DONGHO CHAE 1, RENHUI WAN 2 AND JIAHONG WU 3 arxiv:144.486v2 [math.ap] 3 Jan 215 Abstract. The Hall-magnetohydrodynamics
More informationarxiv: v2 [math.ap] 6 Sep 2007
ON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1, arxiv:0708.3067v2 [math.ap] 6 Sep 2007 A. CHESKIDOV AND R. SHVYDKOY ABSTRACT. We show that if a Leray-Hopf solution u to the
More informationJournal of Differential Equations
J. Differential Equations 48 (1) 6 74 Contents lists available at ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde Two regularity criteria for the D MHD equations Chongsheng
More informationOSGOOD TYPE REGULARITY CRITERION FOR THE 3D NEWTON-BOUSSINESQ EQUATION
Electronic Journal of Differential Equations, Vol. 013 (013), No. 3, pp. 1 6. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu OSGOOD TYPE REGULARITY
More informationOn the local existence for an active scalar equation in critical regularity setting
On the local existence for an active scalar equation in critical regularity setting Walter Rusin Department of Mathematics, Oklahoma State University, Stillwater, OK 7478 Fei Wang Department of Mathematics,
More informationCOMPONENT REDUCTION FOR REGULARITY CRITERIA OF THE THREE-DIMENSIONAL MAGNETOHYDRODYNAMICS SYSTEMS
Electronic Journal of Differential Equations, Vol. 4 4, No. 98,. 8. ISSN: 7-669. UR: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu ft ejde.math.txstate.edu COMPONENT REDUCTION FOR REGUARITY CRITERIA
More informationGlobal well-posedness of the primitive equations of oceanic and atmospheric dynamics
Global well-posedness of the primitive equations of oceanic and atmospheric dynamics Jinkai Li Department of Mathematics The Chinese University of Hong Kong Dynamics of Small Scales in Fluids ICERM, Feb
More informationON THE EXISTENCE AND SMOOTHNESS PROBLEM OF THE MAGNETOHYDRODYNAMICS SYSTEM KAZUO YAMAZAKI
ON THE EXISTENCE AND SMOOTHNESS PROBLEM OF THE MAGNETOHYDRODYNAMICS SYSTEM By KAZUO YAMAZAKI Bachelor of Arts in Political Science City University of New York Brooklyn, New York, U.S.A. 26 Master of Science
More informationGLOBAL REGULARITY RESULTS FOR THE CLIMATE MODEL WITH FRACTIONAL DISSIPATION
GOBA REGUARITY RESUTS FOR THE CIMATE MODE WITH FRACTIONA DISSIPATION BO-QING DONG 1, WENJUAN WANG 1, JIAHONG WU AND HUI ZHANG 3 Abstract. This paper studies the global well-posedness problem on a tropical
More informationLocal Well-Posedness for the Hall-MHD Equations with Fractional Magnetic Diffusion
J. Math. Fluid Mech. 17 (15), 67 638 c 15 Springer Basel 14-698/15/467-1 DOI 1.17/s1-15--9 Journal of Mathematical Fluid Mechanics Local Well-Posedness for the Hall-MHD Equations with Fractional Magnetic
More informationA New Regularity Criterion for the 3D Navier-Stokes Equations via Two Entries of the Velocity Gradient
Acta Appl Math (014) 19:175 181 DOI 10.1007/s10440-013-9834-3 A New Regularity Criterion for the 3D Navier-Stokes Euations via Two Entries of the Velocity Gradient Tensor Zujin Zhang Dingxing Zhong Lin
More informationarxiv: v1 [math.ap] 14 Nov 2016
ON THE INITIAL- AN BOUNARY-VALUE PROBLEM FOR 2 MICROPOLAR EQUATIONS WITH ONLY ANGULAR VELOCITY ISSIPATION arxiv:1611.4236v1 [math.ap] 14 Nov 216 QUANSEN JIU, JITAO LIU, JIAHONG WU, HUAN YU Abstract. This
More informationDissipative quasi-geostrophic equations with L p data
Electronic Journal of Differential Equations, Vol. (), No. 56, pp. 3. ISSN: 7-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Dissipative quasi-geostrophic
More informationDIRECTION OF VORTICITY AND A REFINED BLOW-UP CRITERION FOR THE NAVIER-STOKES EQUATIONS WITH FRACTIONAL LAPLACIAN
DIRECTION OF VORTICITY AND A REFINED BLOW-UP CRITERION FOR THE NAVIER-STOKES EQUATIONS WITH FRACTIONAL LAPLACIAN KENGO NAKAI Abstract. We give a refined blow-up criterion for solutions of the D Navier-
More informationESTIMATES OF LOWER ORDER DERIVATIVES OF VISCOUS FLUID FLOW PAST A ROTATING OBSTACLE
REGULARITY AND OTHER ASPECTS OF THE NAVIER STOKES EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 7 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 25 ESTIMATES OF LOWER ORDER DERIVATIVES OF
More informationA LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE. 1.
A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE THOMAS CHEN AND NATAŠA PAVLOVIĆ Abstract. We prove a Beale-Kato-Majda criterion
More informationON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1
ON THE REGULARITY OF WEAK SOLUTIONS OF THE 3D NAVIER-STOKES EQUATIONS IN B 1, A. CHESKIDOV AND R. SHVYDKOY ABSTRACT. We show that if a Leray-Hopf solution u to the 3D Navier- Stokes equation belongs to
More informationRemarks on the blow-up criterion of the 3D Euler equations
Remarks on the blow-up criterion of the 3D Euler equations Dongho Chae Department of Mathematics Sungkyunkwan University Suwon 44-746, Korea e-mail : chae@skku.edu Abstract In this note we prove that the
More informationREGULARITY OF GENERALIZED NAVEIR-STOKES EQUATIONS IN TERMS OF DIRECTION OF THE VELOCITY
Electronic Journal of Differential Equations, Vol. 00(00), No. 05, pp. 5. ISSN: 07-669. UR: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu REGUARITY OF GENERAIZED NAVEIR-STOKES
More informationA regularity criterion for the generalized Hall-MHD system
Gu et al. Boundary Value Problems (016 016:188 DOI 10.1186/s13661-016-0695-3 R E S E A R C H Open Access A regularity criterion for the generalized Hall-MHD system Weijiang Gu 1, Caochuan Ma,3* and Jianzhu
More informationSufficient conditions on Liouville type theorems for the 3D steady Navier-Stokes equations
arxiv:1805.07v1 [math.ap] 6 May 018 Sufficient conditions on Liouville type theorems for the D steady Navier-Stokes euations G. Seregin, W. Wang May 8, 018 Abstract Our aim is to prove Liouville type theorems
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationON THE STRONG SOLUTIONS OF THE INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN
ON THE STRONG SOLUTIONS OF THE INHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN A THIN DOMAIN XIAN LIAO Abstract. In this work we will show the global existence of the strong solutions of the inhomogeneous
More informationDecay in Time of Incompressible Flows
J. math. fluid mech. 5 (23) 231 244 1422-6928/3/3231-14 c 23 Birkhäuser Verlag, Basel DOI 1.17/s21-3-79-1 Journal of Mathematical Fluid Mechanics Decay in Time of Incompressible Flows Heinz-Otto Kreiss,
More informationGlobal regularity of a modified Navier-Stokes equation
Global regularity of a modified Navier-Stokes equation Tobias Grafke, Rainer Grauer and Thomas C. Sideris Institut für Theoretische Physik I, Ruhr-Universität Bochum, Germany Department of Mathematics,
More informationarxiv: v1 [math.ap] 12 Mar 2009
LIMITING FRACTIONAL AND LORENTZ SPACES ESTIMATES OF DIFFERENTIAL FORMS JEAN VAN SCHAFTINGEN arxiv:0903.282v [math.ap] 2 Mar 2009 Abstract. We obtain estimates in Besov, Lizorkin-Triebel and Lorentz spaces
More informationFrequency Localized Regularity Criteria for the 3D Navier Stokes Equations. Z. Bradshaw & Z. Grujić. Archive for Rational Mechanics and Analysis
Frequency Localized Regularity Criteria for the 3D Navier Stokes Equations Z. Bradshaw & Z. Gruić Archive for Rational Mechanics and Analysis ISSN 0003-9527 Arch Rational Mech Anal DOI 10.1007/s00205-016-1069-9
More informationhal , version 6-26 Dec 2012
ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS ABDEHAFID YOUNSI Abstract. In this paper, we give a new regularity criterion on the uniqueness results of weak solutions for the 3D Navier-Stokes equations
More informationWELL-POSEDNESS OF THE TWO-DIMENSIONAL GENERALIZED BENJAMIN-BONA-MAHONY EQUATION ON THE UPPER HALF PLANE
WELL-POSEDNESS OF THE TWO-DIMENSIONAL GENERALIZED BENJAMIN-BONA-MAHONY EQUATION ON THE UPPER HALF PLANE YING-CHIEH LIN, C. H. ARTHUR CHENG, JOHN M. HONG, JIAHONG WU, AND JUAN-MING YUAN Abstract. This paper
More informationASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp. 189 04 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of
More informationVANISHING VISCOSITY LIMIT FOR THE 3D NONHOMOGENEOUS INCOMPRESSIBLE NAVIER-STOKES EQUATION WITH SPECIAL SLIP BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 169, pp. 1 13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu VANISHING VISCOSITY LIMIT FOR THE 3D NONHOMOGENEOUS
More informationVANISHING VISCOSITY IN THE PLANE FOR NONDECAYING VELOCITY AND VORTICITY
VANISHING VISCOSITY IN THE PLANE FOR NONDECAYING VELOCITY AND VORTICITY ELAINE COZZI Abstract. Assuming that initial velocity and initial vorticity are bounded in the plane, we show that on a sufficiently
More informationUNIFORM DECAY OF SOLUTIONS FOR COUPLED VISCOELASTIC WAVE EQUATIONS
Electronic Journal of Differential Equations, Vol. 16 16, No. 7, pp. 1 11. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu UNIFORM DECAY OF SOLUTIONS
More informationParaproducts and the bilinear Calderón-Zygmund theory
Paraproducts and the bilinear Calderón-Zygmund theory Diego Maldonado Department of Mathematics Kansas State University Manhattan, KS 66506 12th New Mexico Analysis Seminar April 23-25, 2009 Outline of
More informationResearch Statement. 1 Overview. Zachary Bradshaw. October 20, 2016
Research Statement Zachary Bradshaw October 20, 2016 1 Overview My research is in the field of partial differential equations. I am primarily interested in the three dimensional non-stationary Navier-Stokes
More informationWeek 6 Notes, Math 865, Tanveer
Week 6 Notes, Math 865, Tanveer. Energy Methods for Euler and Navier-Stokes Equation We will consider this week basic energy estimates. These are estimates on the L 2 spatial norms of the solution u(x,
More informationOn the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in R 3
Z. Angew. Math. Phys. 016 67:18 c 016 The Authors. This article is published with open access at Springerlink.com 0044-75/16/00001-10 published online April, 016 DOI 10.1007/s000-016-0617- Zeitschrift
More informationThe Role of Convection and Nearly Singular Behavior of the 3D Navier-Stokes Equations
The Role of Convection and Nearly Singular Behavior of the 3D Navier-Stokes Equations Thomas Y. Hou Applied and Comput. Mathematics, Caltech PDE Conference in honor of Blake Temple, University of Michigan
More informationResearch Article Remarks on the Regularity Criterion of the Navier-Stokes Equations with Nonlinear Damping
Mathematical Problems in Engineering Volume 15, Article ID 194, 5 pages http://dx.doi.org/1.1155/15/194 Research Article Remarks on the Regularity Criterion of the Navier-Stokes Equations with Nonlinear
More information*Corresponding Author: Surajit Dutta, Department of Mathematics, C N B College, Bokakhat, Golaghat, Assam, India
International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 6, Issue, 8, PP -6 ISSN 347-37X (Print) & ISSN 347-34 (Online) DOI: http://dx.doi.org/.43/347-34.6 www.arcjournals.org
More informationNew Discretizations of Turbulent Flow Problems
New Discretizations of Turbulent Flow Problems Carolina Cardoso Manica and Songul Kaya Merdan Abstract A suitable discretization for the Zeroth Order Model in Large Eddy Simulation of turbulent flows is
More informationREMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID
REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID DRAGOŞ IFTIMIE AND JAMES P. KELLIHER Abstract. In [Math. Ann. 336 (2006), 449-489] the authors consider the two dimensional
More informationPartial regularity for suitable weak solutions to Navier-Stokes equations
Partial regularity for suitable weak solutions to Navier-Stokes equations Yanqing Wang Capital Normal University Joint work with: Quansen Jiu, Gang Wu Contents 1 What is the partial regularity? 2 Review
More informationGLOBAL WELL-POSEDNESS IN SPATIALLY CRITICAL BESOV SPACE FOR THE BOLTZMANN EQUATION
GLOBAL WELL-POSEDNESS IN SPATIALLY CRITICAL BESOV SPACE FOR THE BOLTZMANN EQUATION RENJUN DUAN, SHUANGQIAN LIU, AND JIANG XU Abstract. The unique global strong solution in the Chemin-Lerner type space
More informationLOCAL WELL-POSEDNESS FOR AN ERICKSEN-LESLIE S PARABOLIC-HYPERBOLIC COMPRESSIBLE NON-ISOTHERMAL MODEL FOR LIQUID CRYSTALS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 3, pp. 1 8. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LOCAL WELL-POSEDNESS FOR AN ERICKSEN-LESLIE S
More informationREGULARITY RESULTS ON THE LERAY-ALPHA MAGNETOHYDRODYNAMICS SYSTEMS
REGULARITY RESULTS ON THE LERAY-ALPHA MAGNETOHYDRODYNAMICS SYSTEMS DURGA KC AND KAZUO YAMAZAKI Abstract. We study certain generalized Leray-alha magnetohydrodynamics systems. We show that the solution
More informationGlobal Well-posedness of the Incompressible Magnetohydrodynamics
Global Well-posedness of the Incompressible Magnetohydrodynamics arxiv:165.439v1 [math.ap] May 16 Yuan Cai Zhen Lei Abstract This paper studies the Cauchy problem of the incompressible magnetohydrodynamic
More informationBlow up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation
Blow up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation Dong Li a,1 a School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 854,
More informationOn a Suitable Weak Solution of the Navier Stokes Equation with the Generalized Impermeability Boundary Conditions
Proceedings of the 3rd IASME/WSEAS Int. Conf. on FLUID DYNAMICS & AERODYNAMICS, Corfu, Greece, August -, 5 pp36-41 On a Suitable Weak Solution of the Navier Stokes Equation with the Generalized Impermeability
More informationA regularity criterion for the 3D NSE in a local version of the space of functions of bounded mean oscillations
Ann. I. H. Poincaré AN 27 (2010) 773 778 www.elsevier.com/locate/anihpc A regularity criterion for the 3D NSE in a local version of the space of functions of bounded mean oscillations Zoran Grujić a,,
More informationOn Global Well-Posedness of the Lagrangian Averaged Euler Equations
On Global Well-Posedness of the Lagrangian Averaged Euler Equations Thomas Y. Hou Congming Li Abstract We study the global well-posedness of the Lagrangian averaged Euler equations in three dimensions.
More informationOPTIMAL CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH POTENTIAL FORCES
OPTIMAL CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH POTENTIAL FORCES RENJUN DUAN Department of Mathematics, City University of Hong Kong 83 Tat Chee Avenue, Kowloon, Hong Kong,
More informationINSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES. Note on the fast decay property of steady Navier-Stokes flows in the whole space
INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES Note on the fast decay property of stea Navier-Stokes flows in the whole space Tomoyuki Nakatsuka Preprint No. 15-017 PRAHA 017 Note on the fast
More informationFENG CHENG, WEI-XI LI, AND CHAO-JIANG XU
GEVERY REGULARITY WITH WEIGHT FOR INCOMPRESSIBLE EULER EQUATION IN THE HALF PLANE arxiv:151100539v2 [mathap] 23 Nov 2016 FENG CHENG WEI-XI LI AND CHAO-JIANG XU Abstract In this work we prove the weighted
More informationRemarks on the Method of Modulus of Continuity and the Modified Dissipative Porous Media Equation
Remarks on the Method of Modulus of Continuity and the Modified Dissipative Porous Media Equation Kazuo Yamazaki 1 2 Abstract We employ Besov space techniques and the method of modulus of continuity to
More informationSome asymptotic properties of solutions for Burgers equation in L p (R)
ARMA manuscript No. (will be inserted by the editor) Some asymptotic properties of solutions for Burgers equation in L p (R) PAULO R. ZINGANO Abstract We discuss time asymptotic properties of solutions
More informationParash Moni Thakur. Gopal Ch. Hazarika
International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 2, Issue 6, June 2014, PP 554-566 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org Effects of
More informationVANISHING VISCOSITY IN THE PLANE FOR VORTICITY IN BORDERLINE SPACES OF BESOV TYPE
VANISHING VISCOSITY IN THE PLANE FOR VORTICITY IN BORDERLINE SPACES OF BESOV TYPE ELAINE COZZI AND JAMES P. KELLIHER Abstract. The existence and uniqueness of solutions to the Euler equations for initial
More informationON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS
ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS Abdelhafid Younsi To cite this version: Abdelhafid Younsi. ON THE UNIQUENESS IN THE 3D NAVIER-STOKES EQUATIONS. 4 pages. 212. HAL Id:
More informationAvailable online at J. Math. Comput. Sci. 4 (2014), No. 3, ISSN:
Available online at http://scik.org J. Math. Comput. Sci. 4 (2014), No. 3, 587-593 ISSN: 1927-5307 A SMALLNESS REGULARITY CRITERION FOR THE 3D NAVIER-STOKES EQUATIONS IN THE LARGEST CLASS ZUJIN ZHANG School
More informationDISSIPATIVE MODELS GENERALIZING THE 2D NAVIER-STOKES AND THE SURFACE QUASI-GEOSTROPHIC EQUATIONS
DISSIPATIVE MODELS GENERALIZING THE 2D NAVIER-STOKES AND THE SURFACE QUASI-GEOSTROPHIC EQUATIONS DONGHO CHAE, PETER CONSTANTIN 2 AND JIAHONG WU 3 Abstract. This paper is devoted to the global (in time)
More informationBlow-up or No Blow-up? the Role of Convection in 3D Incompressible Navier-Stokes Equations
Blow-up or No Blow-up? the Role of Convection in 3D Incompressible Navier-Stokes Equations Thomas Y. Hou Applied and Comput. Mathematics, Caltech Joint work with Zhen Lei; Congming Li, Ruo Li, and Guo
More informationOn the Regularity of Weak Solutions to the Magnetohydrodynamic Equations
On the Regularity of Weak Solutions to the Magnetohydrodynamic Equations Cheng HE (Institute of Applied Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, 18,
More informationLORENTZ SPACE ESTIMATES FOR VECTOR FIELDS WITH DIVERGENCE AND CURL IN HARDY SPACES
- TAMKANG JOURNAL OF MATHEMATICS Volume 47, Number 2, 249-260, June 2016 doi:10.5556/j.tkjm.47.2016.1932 This paper is available online at http://journals.math.tku.edu.tw/index.php/tkjm/pages/view/onlinefirst
More informationA generalised Ladyzhenskaya inequality and a coupled parabolic-elliptic problem
A generalised Ladyzhenskaya inequality and a coupled parabolic-elliptic problem Dave McCormick joint work with James Robinson and José Rodrigo Mathematics and Statistics Centre for Doctoral Training University
More informationLARGE TIME BEHAVIOR OF THE RELATIVISTIC VLASOV MAXWELL SYSTEM IN LOW SPACE DIMENSION
Differential Integral Equations Volume 3, Numbers 1- (1), 61 77 LARGE TIME BEHAVIOR OF THE RELATIVISTIC VLASOV MAXWELL SYSTEM IN LOW SPACE DIMENSION Robert Glassey Department of Mathematics, Indiana University
More informationControl of Interface Evolution in Multi-Phase Fluid Flows
Control of Interface Evolution in Multi-Phase Fluid Flows Markus Klein Department of Mathematics University of Tübingen Workshop on Numerical Methods for Optimal Control and Inverse Problems Garching,
More informationOptimal L p (1 p ) rates of decay to linear diffusion waves for nonlinear evolution equations with ellipticity and dissipation
Nonlinear Analysis ( ) www.elsevier.com/locate/na Optimal L p (1 p ) rates of decay to linear diffusion waves for nonlinear evolution equations with ellipticity and dissipation Renjun Duan a,saipanlin
More informationStability for a class of nonlinear pseudo-differential equations
Stability for a class of nonlinear pseudo-differential equations Michael Frankel Department of Mathematics, Indiana University - Purdue University Indianapolis Indianapolis, IN 46202-3216, USA Victor Roytburd
More informationA new regularity criterion for weak solutions to the Navier-Stokes equations
A new regularity criterion for weak solutions to the Navier-Stokes equations Yong Zhou Department of Mathematics, East China Normal University Shanghai 6, CHINA yzhou@math.ecnu.edu.cn Proposed running
More informationMiami, Florida, USA. Engineering, University of California, Irvine, California, USA. Science, Rehovot, Israel
This article was downloaded by:[weizmann Institute Science] On: July 008 Access Details: [subscription number 7918096] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered
More informationERROR ANALYSIS OF STABILIZED SEMI-IMPLICIT METHOD OF ALLEN-CAHN EQUATION. Xiaofeng Yang. (Communicated by Jie Shen)
DISCRETE AND CONTINUOUS doi:1.3934/dcdsb.29.11.157 DYNAMICAL SYSTEMS SERIES B Volume 11, Number 4, June 29 pp. 157 17 ERROR ANALYSIS OF STABILIZED SEMI-IMPLICIT METHOD OF ALLEN-CAHN EQUATION Xiaofeng Yang
More informationTHE EXISTENCE OF GLOBAL ATTRACTOR FOR A SIXTH-ORDER PHASE-FIELD EQUATION IN H k SPACE
U.P.B. Sci. Bull., Series A, Vol. 8, Iss. 2, 218 ISSN 1223-727 THE EXISTENCE OF GLOBAL ATTRACTOR FOR A SIXTH-ORDER PHASE-FIELD EQUATION IN H k SPACE Xi Bao 1, Ning Duan 2, Xiaopeng Zhao 3 In this paper,
More informationDecay rate of the compressible Navier-Stokes equations with density-dependent viscosity coefficients
South Asian Journal of Mathematics 2012, Vol. 2 2): 148 153 www.sajm-online.com ISSN 2251-1512 RESEARCH ARTICLE Decay rate of the compressible Navier-Stokes equations with density-dependent viscosity coefficients
More informationTHE POINCARÉ RECURRENCE PROBLEM OF INVISCID INCOMPRESSIBLE FLUIDS
ASIAN J. MATH. c 2009 International Press Vol. 13, No. 1, pp. 007 014, March 2009 002 THE POINCARÉ RECURRENCE PROBLEM OF INVISCID INCOMPRESSIBLE FLUIDS Y. CHARLES LI Abstract. Nadirashvili presented a
More informationBesov-type spaces with variable smoothness and integrability
Besov-type spaces with variable smoothness and integrability Douadi Drihem M sila University, Department of Mathematics, Laboratory of Functional Analysis and Geometry of Spaces December 2015 M sila, Algeria
More informationarxiv:math/ v1 [math.ap] 16 Nov 2006
arxiv:math/611494v1 [mathap] 16 Nov 26 ON THE GLOBAL SOLUTIONS OF THE SUPER-CRITICAL 2D QUASI-GEOSTROPHIC EQUATION IN BESOV SPACES TAOUFIK HMIDI AND SAHBI KERAANI Abstract In this paper we study the super-critical
More informationBLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION EQUATION WITH HOMOGENEOUS NEUMANN BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 016 (016), No. 36, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST
More informationEXISTENCE AND REGULARITY OF SOLUTIONS FOR STOKES SYSTEMS WITH NON-SMOOTH BOUNDARY DATA IN A POLYHEDRON
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 147, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND REGULARITY OF SOLUTIONS FOR
More informationOn the Stokes operator in general unbounded domains. Reinhard Farwig, Hideo Kozono and Hermann Sohr
Hokkaido Mathematical Journal Vol. 38 (2009) p. 111 136 On the Stokes operator in general unbounded domains Reinhard Farwig, Hideo Kozono and Hermann Sohr (Received June 27, 2007; Revised December 19,
More informationEXISTENCE AND UNIQUENESS FOR A THREE DIMENSIONAL MODEL OF FERROMAGNETISM
1 EXISTENCE AND UNIQUENESS FOR A THREE DIMENSIONAL MODEL OF FERROMAGNETISM V. BERTI and M. FABRIZIO Dipartimento di Matematica, Università degli Studi di Bologna, P.zza di Porta S. Donato 5, I-4126, Bologna,
More informationOn Behaviors of the Energy of Solutions for Some Damped Nonlinear Hyperbolic Equations with p-laplacian Soufiane Mokeddem
International Journal of Advanced Research in Mathematics ubmitted: 16-8-4 IN: 97-613, Vol. 6, pp 13- Revised: 16-9-7 doi:1.185/www.scipress.com/ijarm.6.13 Accepted: 16-9-8 16 cipress Ltd., witzerland
More informationThe incompressible Navier-Stokes equations in vacuum
The incompressible, Université Paris-Est Créteil Piotr Bogus law Mucha, Warsaw University Journées Jeunes EDPistes 218, Institut Elie Cartan, Université de Lorraine March 23th, 218 Incompressible Navier-Stokes
More informationhal , version 1-22 Nov 2009
Author manuscript, published in "Kinet. Relat. Models 1, 3 8) 355-368" PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS HUA CHEN, WEI-XI LI AND CHAO-JIANG XU Abstract. By using the energy-type
More informationThe Navier-Stokes Equations with Time Delay. Werner Varnhorn. Faculty of Mathematics University of Kassel, Germany
The Navier-Stokes Equations with Time Delay Werner Varnhorn Faculty of Mathematics University of Kassel, Germany AMS: 35 (A 35, D 5, K 55, Q 1), 65 M 1, 76 D 5 Abstract In the present paper we use a time
More informationThe 3D Euler and 2D surface quasi-geostrophic equations
The 3D Euler and 2D surface quasi-geostrophic equations organized by Peter Constantin, Diego Cordoba, and Jiahong Wu Workshop Summary This workshop focused on the fundamental problem of whether classical
More informationA global solution curve for a class of free boundary value problems arising in plasma physics
A global solution curve for a class of free boundary value problems arising in plasma physics Philip Korman epartment of Mathematical Sciences University of Cincinnati Cincinnati Ohio 4522-0025 Abstract
More informationISABELLE GALLAGHER AND MARIUS PAICU
REMARKS ON THE BLOW-UP OF SOLUTIONS TO A TOY MODEL FOR THE NAVIER-STOKES EQUATIONS ISABELLE GALLAGHER AND MARIUS PAICU Abstract. In [14], S. Montgomery-Smith provides a one dimensional model for the three
More informationLocal null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain
Local null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain Nicolás Carreño Université Pierre et Marie Curie-Paris 6 UMR 7598 Laboratoire
More informationMath 497 R1 Winter 2018 Navier-Stokes Regularity
Math 497 R Winter 208 Navier-Stokes Regularity Lecture : Sobolev Spaces and Newtonian Potentials Xinwei Yu Jan. 0, 208 Based on..2 of []. Some ne properties of Sobolev spaces, and basics of Newtonian potentials.
More informationBlow-up or No Blow-up? Fluid Dynamic Perspective of the Clay Millennium Problem
Blow-up or No Blow-up? Fluid Dynamic Perspective of the Clay Millennium Problem Thomas Y. Hou Applied and Comput. Mathematics, Caltech Research was supported by NSF Theodore Y. Wu Lecture in Aerospace,
More information