Forward Discretely Self-Similar Solutions of the MHD Equations and the Viscoelastic Navier-Stokes Equations with Damping
Chen-Chih Lai








Forward Discretely Self-Similar Solutions of the MHD Equations and the Viscoelastic Navier-Stokes Equations with Damping
    Chen-Chih Lai
    
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In this paper, we prove the existence of forward discretely self-similar solutions to the MHD equations and the viscoelastic Navier-Stokes equations with damping with large weak L^3 initial data. The same proving techniques are also applied to construct self-similar solutions to the MHD equations and the viscoelastic Navier-Stokes equations with damping with large weak L^3 initial data. This approach is based on [Z. Bradshaw and T.-P. Tsai, Ann. Henri Poincar'e, vol. 18, no. 3, 1095-1119, 2017].












§ INTRODUCTION

The main purpose of this paper is to prove the existence of forward discretely self-similar (DSS) and self-similar (SS) weak solutions of both the MHD equations and the viscoelastic Navier-Stokes equations with damping. More precisely, we construct DSS local Leray weak solutions for DSS initial data with possibly large L^3_w-norm, and SS local Leray solutions for (-1)-homogeneous initial data in L^3_w. Our method follows from <cit.> and is based on the a priori bounds (<ref>) and (<ref>), and the Galerkin method. To begin with, we briefly introduce the MHD equations and the viscoelastic Navier-Stokes equations.


 §.§ The incompressible MHD equations

In a magnetofluid, the interaction between the velocity field of the fluid and the magnetic field is governed by the coupling between the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. The fundamental equations of magentohydrodynamics (MHD) is given by

    1.2.[ _tv-ν_0 v+(v·)v-(b·)b+π                      =0;   _tb-η_0 b+(v·)b-(b·)v                      =0;                · v =· b                      =0 ]} in ^3×(0,∞),

with initial data 
    v|_t=0=v_0  and  b|_t=0=b_0  in ^3,

where u:^3×(0,∞)→^3 is the fluid velocity, b:^3×(0,∞)→^3 is the magnetic field, and π:^3×(0,∞)→ represents the fluid pressure. The constants ν_0>0 and η_0>0 are the kinetic viscosity and the magnetic resistivity, respectively. For simplicity, we assume ν_0=η_0=1 throughout this paper.

We recall that the MHD equations (<ref>) is invariant under the scaling 

v^(x,t)   =  v(x,^2t), v_0^(x)= v_0(x),

b^(x,t)   =  b(x,^2t),   b_0^(x)= b_0(x),

π^(x,t)   =^2π(x,^2t).

We say that a solution (v,b,π) of (<ref>) is self-similar (SS) if it satisfies the scaling invariant v^=v, b^=b and π^=π for all >0. The initial data v_0 and b_0 are called self-similar if v_0^=v_0 and b_0^=b_0. On the other hand, if the scaling invariant only holds for a particular >0, we say (v,b,π) is discretely self-similar with factor >1 (-DSS). Similarly, the initial data v_0 and b_0 are said to be -DSS if v_0^=v_0 and b_0^=b_0 for this >1.

On one hand, self-similar solutions of (<ref>) have a stationary characteristic in that there exists an ansatz for (v,b) in terms of time-independent profile (u,a). That is, 

    v(x,t)=1/√(2t) u(x/√(2t)),   b(x)=1/√(2t) a(x/√(2t)),   π(x,t)=1/2t p(x/√(2t)).

The profile (u,a) solves the stationary Leray system for the MHD equations

    1.2.[ - u-u-y· u+(u·)u-(a·)a+ p                        =0;    - a-a-y· a+(u·)a-(a·)u                        =0;                  · u =· b                        =0 ]} in ^3×,

in the variable y=x/√(2t). On the other hand, discretely self-similar solutions of (<ref>) are determined by the behavior on the time intervals of the form 1≤ t≤^2. This leads us to consider the self-similar transform 

    v(x,t)=1/√(2t) u(y,s),   b(x,t)=1/√(2t) a(y,s),   π(x,t)=1/2t p(y,s),

where 

    y=x/√(2t),   s=log(√(2t)).

Then (u,a,p) solves the time-dependent Leray system for the MHD equations 

    1.2.[ _su- u-u-y· u+(u·)u-(a·)a+ p                           =0;    _sa- a-a-y· a+(u·)a-(a·)u                           =0;                     · u =· b                           =0 ]} in ^3×.

Note that (v,b,π) is -DSS if and only if (u,a,p) is periodic in s with the period T=log().

Many significant contributions have been made concerning the existence of solutions to the MHD equations (<ref>). We list only some results related to our studies. First, Duvaut and Lions <cit.> constructed a class of global weak solutions with finite energy and a class of local strong solutions. And the unique existence of mild solutions in BMO^-1 for small initial data has been obtained in Miao-Yuan-Zhang <cit.>. In He-Xin <cit.>, they also constructed a class of global unique forward SS solutions for small (-1)-homogeneous initial data belonging to some Besov space, or the Lorentz space or pseudo-measure space. Recently, Lin-Zhang-Zhou <cit.> constructed a class of global smooth solution for large initial data assuming some constraints on the initial data on Fourier side.




 §.§ The incompressible viscoelastic Navier-Stokes equations with damping

The Oldroyd-type models capture the rheological phenomena of both the fluid motions and the elastic features of non-Newtonian fluids. We study the simplest case in which the relaxation and retardation times are both infinite. More specifically, we consider the following system of equations for an incompressible, viscoelastic fluid: 

    1.2.[ _tv-ν_0 v+(v·)v-·( F F^⊤)+π                          =0;          _t F+(v·) F-( v) F                          =0;                         · v                          =0 ]} in ^3×(0,∞),

with initial data 
    v|_t=0=v_0  and   F|_t=0= F_0  in ^3,

where u:^3×(0,∞)→^3 is the velocity field, F:^3×(0,∞)→^3×3 is the local deformation tensor of the fluid, and π:^3×(0,∞)→ represents the pressure. The constant ν_0>0 is the kinetic viscosity. Here (∇·( F F^⊤))_i=_j( F_ik F_jk) and ( v)_ij=_jv_i. For convenience, we assume ν_0=1 throughout this paper.

For the existence of weak solutions for the viscoelastic Navier-Stokes equations (<ref>), it is well-known that short-time classical solutions and global existence of classical solutions for small initial data were established by Lin-Liu-Zhang <cit.>. Later on, the authors <cit.> proved the global existence of smooth solutions to (<ref>) in the case of near-equilibrium initial data. In <cit.>, the authors added a damping term in the equation for F of the system (<ref>) to overcome the difficulty arises from the lack of a damping mechanism on F. To be more precise, they introduced the following viscoelastic Navier-Stokes equations with damping as a way to approximate solutions of (<ref>):

    1.2.[ _tv- v+(v·)v-·( F F^⊤)+π                       =0;   _t F-μ F+(v·) F-( v) F                       =0;                      · v                       =0 ]} in ^3×(0,∞),

for a damping parameter μ>0. Note that if · F=0 at some instance of time, then · F=0 at all later times. In fact, by taking divergence of (<ref>)_2 and using (<ref>)_3, one have the following equation for · F:

    _t(· F)+(v·)(· F)=μ(· F).

 Hence it is natural to assume 

·F=0.

Because the damping parameter μ plays no role in our construction of solutions, we set throughout this paper that 
    μ=1.
 Then, columnwisely, (<ref>) can be rewritten as 

    1.2.[ _tv- v+(v·)v-n=13∑(f_n·)f_n+π                            =0;    _tf_m- f_m+(v·)f_m-(f_m·)v                            =0;                     · f_m=· v                            =0 ]} in ^3×(0,∞), m=1,2,3,

where f_m is the m-th column vector of F.

Similar to the MHD equations, the viscoelastic equations with damping (<ref>) is invariant under the scaling 

v^(x,t)   =  v(x,^2t), v_0^(x)= v_0(x),

F^(x,t)   =  F(x,^2t),   F_0^(x)= F_0(x),

π^(x,t)   =^2π(x,^2t).
 
We define SS and -DSS solution to (<ref>) in the same manner as the ones we defined for the MHD equations. Self-similar solutions of (<ref>) is determined by time-periodic profile (u, F), where

    v(x,t)=1/√(2t) u(x/√(2t)),    F(x)=1/√(2t)  G(x/√(2t)),   π(x,t)=1/2t p(x/√(2t)),

which satisfy the stationary Leray system for the viscoelastic Navier-Stokes equations with damping

    1.2.[ - u-u-y· u+(u·)u-n=13∑(g_n·)g_n+ p                                 =0;   - g_m-g_m-y· g_m+(u·)g_m-(g_m·)u                                 =0;                         · u =· g_m                                 =0 ]} in ^3×, m=1,2,3,

where g_m is the m-th column vector of G. For discretely self-similar solutions of (<ref>), we consider the self-similar transform 

    v(x,t)=1/√(2t) u(y,s),    F(x,t)=1/√(2t)  G(y,s),   π(x,t)=1/2t p(y,s),

where x,t,y,s satisfy (<ref>).
Then (u, G,p) solves the time-dependent Leray system for the viscoelastic Navier-Stokes equations with damping

    1.2.[ _su- u-u-y· u+(u·)u-n=13∑(g_n·)g_n+ p                                    =0; _sg_m- g_m-g_m-y· g_m+(u·)g_m-(g_m·)u                                    =0;                            · u =· g_m                                    =0 ]} in ^3×, m=1,2,3,

where g_m is the m-th column vector of G. Note that (v, F,π) is -DSS if and only if (u, G,p) is periodic in s with the period T=log().

The authors <cit.> mentioned that passing the limit of solutions to (<ref>) as μ→0^+ throughout standard weak convergence methods is not able to get weak solutions of (<ref>). Despite of that, (<ref>) itself is still an interesting system, and there are a few of studies on this system. For instance, Lai-Lin-Wang <cit.> established the existence of global forward SS classical solution to (<ref>) for locally Hölder continuous, (-1)-homogeneous initial data. For regularity issues, we refer the reader to <cit.> and <cit.>.




 §.§ Main results and Notation

Our first goal is to extend the notion of weak solutions to the ones with a more general initial data. To this end, we recall the definition of local Leray weak solutions of the MHD equations (<ref>), which is consistent with the concept introduced by Lemarié-Rieusset <cit.> on the Navier-Stokes equations. Here, for 1≤ q<∞, let L^q_ denote the space of functions in ^3 with 

    f_L^q_:=sup_x_0∈^3f_L^q(B_1(x_0))<∞.

def_loc_leray_mhd
A pair of vector fields (v,b), where v,b:^3×[0,∞)→^3 and v,b∈ L^2_(^3×[0,∞)), is called a local Leray solution to (<ref>) with divergence-free initial data v_0, b_0∈ L_^2 if 


  * there exists π∈ L_^3/2(^3×[0,∞)) such that (v,b,π) is a distributional solution to (<ref>),

  * (Locally finite energy/enstrophy) for any R>0, (v,b) satisfies 

   0≤t<R^2 x_0∈^3sup∫_B_R(x_0)1/2(|v(x,t)|^2+|b(x,t)|^2)dx

                           +x_0∈^3sup∫_0^R^2∫_B_R(x_0)(|v(x,t)|^2+|b(x,t)|^2)dxdt<∞,


  * (Decay at spatial infinity) for any R>0, (v,b) satisfies 

lim_|x_0|→∞∫_0^R^2∫_B_R(x_0)(|v(x,t)|^2+|b(x,t)|^2)dxdt=0,


  * (Convergence to initial data) for all compact subsets K of ^3 we have v(t)→ v_0 and b(t)→ b_0 in L^2(K) as t→0^+,

  * (Local energy inequality) for all cylinders Q compactly contained in ^3×(0,∞) and all nonnegetive ϕ∈ C^∞_0(Q), we have 

2∫∫(|v|^2+|b|^2)ϕ dxdt≤   ∫∫(|v|^2+|b|^2)(_tϕ+ϕ)dxdt
   +∫∫(|v|^2+|b|^2+2π)(v·ϕ)dxdt
   -2∫∫(v·b)(b·ϕ)dxdt.




One of our goals in this paper is to prove the following existence theorem of a class of forward discretely self-similar solutions of the MHD equations (<ref>).
thm_1.2_mhd
Let v_0 and b_0 be divergence-free, -DSS vector fields for some >1 and satisfy 

v_0_L^3_w(^3)≤c_0,   b_0_L^3_w(^3)≤c_0,

for some constant c_0>0. Then there exists a -DSS local Leray solution (v,b) to (<ref>). Moreover, there exists C_0=C_0(v_0,b_0) so that 

    v(t)-e^tv_0_L^2(^3)≤ C_0 t^1/4,   b(t)-e^tb_0_L^2(^3)≤ C_0 t^1/4
 for any t∈(0,∞).


Also, self-similar solutions of the MHD equations (<ref>) can be constructed with (-1)-homogeneous initial data. Namely, we have
thm_1.3_mhd
Let v_0 and b_0 be divergence-free, (-1)-homogeneous and satisfy (<ref>) for some constant c_0>0. Then there exists a self-similar local Leray solution (v,b) to (<ref>). In addition, there exists C_0=C_0(v_0,b_0) such that 
    v(t)-e^tv_0_L^2(^3)≤ C_0 t^1/4,   b(t)-e^tb_0_L^2(^3)≤ C_0 t^1/4
 for any t∈(0,∞).







We would like to show similar results to thm_1.2_mhd and thm_1.3_mhd for the viscoelastic Navier-Stokes equations with damping (<ref>). For this purpose, we define analogous local Leray solutions to the viscoelastic Navier-Stokes equations with damping (<ref>) as follows.

A pair of a vector field and a tensor field (v, F), where u:^3×(0,∞)→^3, F:^3×(0,∞)→^3×3 and v,f_m∈ L^2_(^3×[0,∞)) for m=1,2,3 with f_m being the m-th column of F, is called a local Leray solution to (<ref>) with divergence-free initial data v_0,  F_0∈ L_^2 if 


  * there exists π∈ L_^3/2(^3×[0,∞)) such that (v, F,π) is a distributional solution to (<ref>),

  * (Locally finite energy/enstrophy) for any R>0, (v, F) satisfies 

   0≤t<R^2 x_0∈^3sup∫_B_R(x_0)1/2(|v(x,t)|^2+|F(x,t)|^2)dx

                         +x_0∈^3sup∫_0^R^2∫_B_R(x_0)(|v(x,t)|^2+|F(x,t)|^2)dxdt<∞,


  * (Decay at spatial infinity) for any R>0, (v, F) satisfies 

lim_|x_0|→∞∫_0^R^2∫_B_R(x_0)(|v(x,t)|^2+|F(x,t)|^2)dxdt=0,


  * (Convergence to initial data) for all compact subsets K of ^3 we have v(t)→ v_0 and F(t)→ F_0 in L^2(K) as t→0^+,

  * (Local energy inequality) for all cylinders Q compactly contained in ^3×(0,∞) and all nonnegative ϕ∈ C^∞_0(Q), we have 

2∫∫(|v|^2+|F|^2)ϕ dxdt≤   ∫∫(|v|^2+|F|^2)(_tϕ+ϕ)dxdt
   +∫∫(|v|^2+|F|^2+2π)(v·ϕ)dxdt
   -2 n=13∑∫∫(v·f_n)(f_n·ϕ)dxdt.




The main theorems in this paper for the viscoelastic Navier-Stokes equations with damping can be stated as the following:
thm_1.2_vNSEd
Let v_0 and F_0 be divergence-free, -DSS vector fields for some >1 and satisfy 

v_0_L^3_w(^3)≤c_0,   F_0_L^3_w(^3)≤c_0,

for some constant c_0>0. Then there exists a local Leray solution (v, F) to (<ref>) which is -DSS. Moreover, there exists C_0=C_0(v_0, F_0) so that 

    v(t)-e^tv_0_L^2(^3)≤ C_0 t^1/4,    F(t)-e^t F_0_L^2(^3)≤ C_0 t^1/4
 for any t∈(0,∞).
 
thm_1.3_vNSEd
Let v_0 and F_0 be divergence-free, (-1)-homogeneous and satisfy (<ref>) for some constant c_0>0. Then there exists a self-similar local Leray solution (v, F) to (<ref>). In addition, there exists C_0=C_0(v_0, F_0) so that 

    v(t)-e^tv_0_L^2(^3)≤ C_0 t^1/4,    F(t)-e^t F_0_L^2(^3)≤ C_0 t^1/4
 for any t∈(0,∞).



The solutions obtained in thm_1.3_mhd and thm_1.3_vNSEd are actually infinitely smooth.


The following a priori bounds are the keys to construct our desired solutions. For the MHD equations, if (u,b) is a solution of (<ref>), then the differences U=u-U_0 and A=a-A_0, where U_0 and A_0 are heat solutions, formally satisfy

       ∫_0^T∫(|U|^2+|A|^2+1/2 |U|^2+1/2 |A|^2)

   =∫_0^T∫[(U·)U·U_0+(U·)A·A_0-(A·)U·A_0-(A·)A·U_0]

       -∫_0^T∫[ℛ_1(U_0,A_0)·U+ℛ_2(U_0,A_0)·A],

where ℛ_1(U_0,A_0) and ℛ_2(U_0,A_0) will be given in (<ref>). Similarly, for the viscoelastic Navier-Stokes equations with damping, if (u,g_1,g_2,g_3) is a solution of (<ref>), then the differences U=u-U_0 and G_m=g_m-G_m,0, m=1,2,3, where U_0 and G_m,0 are heat solutions, formally obey 

       ∫_0^T∫(|U|^2+∑_n=1^3|G_n|^2+1/2 |U|^2+1/2 ∑_n=1^3|G_n|^2)

   =∫_0^T∫[(U·)U·U_0+∑_n=1^3(U·)G_n·G_n,0-∑_n=1^3(G_n·)U·G_n,0-∑_n=1^3(G_n·)G_n·U_0]

       -∫_0^T∫[ℛ_3(U_0,G_1,G_2,G_3)·U+∑_n=1^3ℛ_4(U_0,G_n,0)·G_n],

where ℛ_3(U_0,G_1,G_2,G_3) and ℛ_4(U_0,G_n,0) will be given in (<ref>). Note that all cubic terms are either vanish or cancelled out in both (<ref>) and (<ref>). To control the quadratic terms, we will choose a suitable cutoff to eliminate the possibly large local behavior of U_0, A_0 and G_m,0. See lem_2.5 for more details. 




The rest of this paper is organized as follows. In Sect. 2, we recall some results in <cit.> and construct a time-periodic solution to the Leray system for the MHD equations and the viscoelastic Navier-Stokes equations with damping. In Sect. 3, we recover discretely self-similar local Leray solutions for the MHD equations and the viscoelastic Navier-Stokes equations with damping from the solutions of the corresponding Leray systems obtained in Sect. 2. In Sect. 4, we prove the existence of self-similar local Leray solutions for the MHD equations and the viscoelastic Navier-Stokes equations with damping by constructing steady-state solutions to the Leray system for the MHD equations and the viscoelastic Navier-Stokes equations with damping, respectively.



Notation. We define the following function spaces 
𝒱={f∈C^∞_0(^3;^3):·f=0}, X=𝒱^H^1_0(^3), H=𝒱^L^2(^3).

Let (·,·) be the L^2(^3) inner product, and <·,·> be the dual pairing of H^1 and its dual space H^-1, or that for X and X^*. We denote 
    𝒟_T={φ∈ C^∞(^3×;^3):[    φ=0, φ is periodic in s with period T,; (φ(·,s)) is compact in ^3 for all s∈[0,T) ]}.

We recall the Morrey space 
    M^p,={f∈ L^p_loc:f_M^p,:=sup_x∈^3, r>0[r^-∫_B_r(x)|f|^p]^1/p<∞},
 and the weighted L^2 spaces 

    L^2_-k/2={f∈ L^2:∫_^3|f(x)|^2/(1+|x|)^k dx<∞}.






§ THE TIME-PERIODIC LERAY SYSTEM



 §.§ The time-periodic Leray system for the MHD equations

In this subsection, we study the existence of time-periodic weak solutions to the Leray system for the MHD equations

    1.2.[ _su- u-u-y· u+(u·)u-(a·)a+ p                           =0;    _sa- a-a-y· a+(u·)a-(a·)u                           =0;                     · u =· b                           =0 ]}  in ^3×,
    lim_|y_0|→∞∫_B_1(y_0)(|u(y,s)-U_0(y,s)|^2+|a(y,s)-A_0(y,s)|^2)dy=0  for all s∈,
    
    u(·,s)=u(·,s+T), a(·,s)=a(·,s+T)  in ^3 for all s∈,

for given T-periodic divergence-free vector fields U_0 and A_0. 

We first revisit the assumption for the background vector field U_0 and the corresponding results in <cit.>.

[<cit.> Assumption 2.1]assum_2.1
U_0∈ C^1(^4;^3) is periodic in s with period T>0, divergence-free and satisfies

    _sU_0- U_0-U_0-y· U_0=0,


    U_0∈ L^∞(0,T;L^4∩ L^q(^3)),
  

    U_0∈ L^2_loc(^4),


    _s U_0∈ L^∞(0,T;L^6/5_(^3)),

and 

    sup_s∈[0,T]U_0(s)_L^q(^3∖ B_R)≤Θ(R),

for some q∈(3,∞] and Θ:[0,∞)→[0,∞) such that Θ(R)→0 as R→∞.


For notational simplicity, we define the linear differential operator ℒ by 

    ℒW=_sW- W-W-y· W,
 and so 
    <ℒW,ζ>=(_sW-W-y· W,ζ)+( W,ζ)
 for all ζ∈ C^1_0(^3).

lem_2.5
Fix q∈(3,∞] and suppose U_0 satisfies assum_2.1 for this q. Let Z∈ C^∞(^3) with 0≤ Z≤ 1, Z(x)=1 for |x|>2 and Z(x)=0 for |x|<1. For any ∈(0,1), there exists R_0=R_0(U_0,)≥1 so that if we define 
ξ(y)=Z(y/R_0), and  

    w(y,s)=∫_^3_y 1/4π|y-z| _zξ(z)· U_0(z,s)dz,

then

    W(y,s)=ξ(y)U_0(y,s)+w(y,s)

 has the following properties: locally continuously differentiable in y and s, T-periodic, divergence-free, U_0-W∈ L^∞(0,T;L^2(^3))∩ L^2(0,T;H^1(^3)), and 

    W_L^∞(0,T;L^q(^3))≤,


    W_L^∞(0,T;L^4(^3))≤ c(R_0,U_0),

and 

    ℒW_L^∞(0,T;H^-1(^3))≤ c(R_0,U_0),

where c(R_0,U_0) depends on R_0 and quantities associated with U_0 which are finite by assum_2.1.


lem_3.4
Suppose v_0 satisfies the assumption of thm_1.2_mhd and let x,t,y,s satisfy (<ref>). Then 
    U_0(y,s):=√(2t)(e^tv_0)(x)
 satisfies assum_2.1 with T=log() and any q∈(3,∞].



Similar to the Navier-Stokes counterpart of time-periodic Leray system in <cit.>, we define periodic weak solutions and suitable periodic weak solutions of (<ref>) as follows.

 Let U_0 and A_0 both satisfy assum_2.1. A pair of vector fields (u,a) is a periodic weak solution to (<ref>) if · u=· b=0, 
    u-U_0, a-A_0∈ L^∞(0,T;L^2(^3))∩ L^2(0,T;H^1(^3)),

and 

    ∫_0^T[(u,_sφ)-( u,φ)+(u+y· u-u· u+a· a,φ)]ds=0,


    ∫_0^T[(a,_sφ)-( a,φ)+(a+y· a-u· a+a· u,φ)]ds=0,

holds for all φ∈𝒟_T.



 Let U_0 and A_0 both satisfy assum_2.1. A triple (u,a,p) is a suitable periodic weak solution to (<ref>) if u,a,p are periodic in s with period T, (u,a) is a periodic weak solution to (<ref>), p∈ L^3/2_(^4), (u,a,p) solves (<ref>) in the sense of distributions, and the local energy inequality holds: 

∫_^4(|u|^2+|a|^2/2+|u|^2+|a|^2)ψ dyds≤   ∫_^4|u|^2+|a|^2/2(_sψ+ψ)dyds
   +∫_^4(|u|^2+|a|^2/2(u-y)+pu)·ψ dyds
   -∫_^4(u·a)a·ψ dyds,
 for all nonnegative ψ∈ C^∞_0(^4).


We are now ready to prove the existence of suitable periodic weak solutions of (<ref>). Namely, we have
thm_2.4_mhd
Assume U_0(y,s) and A_0(y,s) both satisfy assum_2.1 with q=10/3. Then (<ref>) has a periodic suitable weak solution (u,a,p) in ^4 with period T.


Fix Z∈ C^∞(^3) with 0≤ Z≤ 1, Z(x)=1 for |x|>2 and Z(x)=0 for |x|<1. 
Applying lem_2.5 with =1/4, one can choose R_0=R_0(U_0,A_0)≥1 such that letting ξ(y)=Z(y/R_0) and setting

    W(y,s)=ξ(y)U_0(y,s)+w(y,s)

and 

    D(y,s)=ξ(y)A_0(y,s)+d(y,s),

where 
 
    w(y,s)=∫_^3_y 1/4π|y-z| _zξ(z)· U_0(z,s)dz

 and
  
    d(y,s)=∫_^3_y 1/4π|y-z| _zξ(z)· A_0(z,s)dz,

 both W and D satisfy the conclusion of lem_2.5.
 
Using the differential operator ℒ defined in (<ref>), the Leray system (<ref>) can be written as 

    1.2{[ ℒu+(u·)u-(a·)a+ p                =0;    ℒa+(u·)a-(a·)u                =0;           · u=· a               =0. ].

We are looking for a solution of the form u=U+W and a=A+D. Then (U,A) must satisfy the perturbed Leray system for the MHD equations

    1.2{[ ℒU+(W+U)· U+U· W-(D+A)· A-A· D+ p                        =-ℛ_1(W,D);    ℒA+(W+U)· A+U· D-(D+A)· U-A· W                        =-ℛ_2(W,D);                           · U=· A                               =0, ].

where 

    1.2{[  ℛ_1(W,D):=ℒW+W· W-D· D; ℛ_2(W,D):=ℒD+W· D-D· W. ].


We first solve the following mollified perturbed Leray system for the MHD equations for (U^,A^,p^) in ^3×[0,T]:

    1.2{[                                                   ℒU^+(W+(η_*U^))· U^+U^· W;                                       -(D+(η_*A^))· A^-A^· D+ p^=-ℛ_1(W,D),;                                                   ℒA^+(W+(η_*U^))· A^+U^· D;                                           -(D+(η_*A^))· U^-A^· W=-ℛ_2(W,D),;                                                                · U^=· A^=0, ].

where η_(y)=^-3η(y/) for some fixed function η∈ C^∞_0(^3) satisfying ∫_^3η dy=1. The weak formulation of (<ref>) is 

    1.2{[                                         d/ds(U^,f)                                                  = -( U^, f)+(U^+y· U^,f)-((η_*U^)· U^-(η_*A^)· A^,f);                                                             -(W· U^+U^· W-D· A^-A^· D,f)-<ℛ_1(W,D),f>;                                         d/ds(A^,f)                                                  = -( A^, f)+(A^+y· A^,f)-((η_*U^)· A^-(η_*A^)· U^,f);                                                             -(W· A^+U^· D-D· U^-A^· W,f)-<ℛ_2(W,D),f> ].

for all f∈𝒱 and a.e. s∈(0,T).

 

Step 1: Construction of a solution to the mollified perturbed Leray system

We use the Galerkin method to construct a solution of (<ref>). Let {h_k}_k∈⊂𝒱 be an orthonormal basis of H. Fixing a natural number k, we search for an approximation solution of the form U^_k(y,s)=∑_i=1^kμ^_ki(s)h_i(y), A^_k(y,s)=∑_i=1^k^_ki(s)h_i(y). We first prove the existence and an a priori estimate for T-periodic solutions μ^_k=(μ^_k1,⋯,μ^_kk), ^_k=(^_k1,⋯,^_kk) to the system of ODEs

    1.2{[                                                                       d/dsμ^_kj =i=1k∑𝒜_ijμ^_ki+i=1k∑ℬ_ij^_ki+i,l=1k∑𝒞^_iljμ^_kiμ^_kl-i,l=1k∑𝒞^_ilj^_ki^_kl+𝒟_j;                                                                        d/ds^_kj                       =i=1k∑ℰ_ijμ^_ki+i=1k∑ℱ_ij^_ki+i,l=1k∑𝒢^_iljμ^_ki^_kl+ℋ_j, ].

for j=1,⋯,k, where 

𝒜_ij   =-(h_i,h_j)+(h_i+y·h_i,h_j)-(h_i·W,h_j)-(W·h_i,h_j),

ℬ_ij   =(h_i·D,h_j)+(D·h_i,h_j),

𝒞^_ilj   =-((η_*h_i)·h_l,h_j),

𝒟_j   =-<ℛ_1(W,D),h_j>,

ℰ_ij   =-(h_i·D,h_j)+(D·h_i,h_j),

ℱ_ij   =-(h_i,h_j)+(h_i+y·h_i,h_j)+(h_i·W,h_j)-(W·h_i,h_j),

𝒢^_ilj   =-((η_*h_i)·h_l,h_j)+((η_*h_l)·h_i,h_j),

ℋ_j   =-<ℛ_2(W,D),h_j>.

Fix k∈. For any U^0,A^0∈(h_1,⋯,h_k), there exist μ^_kj,^_kj∈ H^1(0,T̃), j=1,⋯,k, that uniquely solve (<ref>) with initial data μ^_kj(0)=(U^0,h_j), ^_kj(0)=(A^0,h_j), j=1,⋯,k, for some 0<T̃≤ T. 

We show that T̃=T. To this end, we first derive 
       1/2 d/ds(U^_k_L^2^2+A^_k_L^2^2)+1/2(U^_k_L^2^2+A^_k_L^2^2)+(U^_k_L^2^2+A^_k_L^2^2)

   =-(U^_k·W-D·A^_k-A^_k·D,U^_k)-(U^_k·D-D·U^_k-A^_k·W,A^_k)

      -<ℛ_1(W,D),U^_k>-<ℛ_2(W,D),A^_k>
,
by multiplying the j-th equation of (<ref>)_1 by μ^_kj, and multiplying the j-th equation of (<ref>)_2 by ^_kj, and then sum up all 2k equations. In the derivation, notice that ((η_*U^)· U^,U^), (W· U^,U^), ((η_*U^)· A^,A^) and (W· A^,A^) vanish, and ((η_*A^)· A^,U^) and ((η_*A^)· U^,A^) are cancelled each other; thus these terms don't show up in (<ref>). Using lem_2.5 with =1/4, we get 

       |-(U^_k·W-D·A^_k-A^_k·D,U^_k)-(U^_k·D-D·U^_k-A^_k·W,A^_k)|

   ≤3/8(U^_k_H^1^2+A^_k_H^1^2),

and

|-<ℛ_1(W,D),U^_k>-<ℛ_2(W,D),A^_k>|≤C_2+3/32(U^_k_H^1^2+A^_k_H^1^2),

where C_2=8(ℒW_H^-1^2+ℒD_H^-1^2+(W_L^4^2+D_L^4^2)^2) is independent of s,T,k and ε.

Using the estimates (<ref>) and (<ref>), we obtain from (<ref>) the differential inequality

d/ds(U^_k_L^2^2+A^_k_L^2^2)+1/16(U^_k_L^2^2+A^_k_L^2^2)+1/16(U^_k_L^2^2+A^_k_L^2^2)≤C_2.

Applying the Gronwall inequality, we get 

e^s/16(U^_k_L^2^2+A^_k_L^2^2)≤    (U^0_L^2^2+A^0_L^2^2)+∫_0^T̃e^τ/16C_2 dτ

≤    (U^0_L^2^2+A^0_L^2^2)+e^T/16C_2T

for all s∈[0,T̃]. Since the right-hand side is finite, T̃ is not a blow-up time and we conclude that T̃=T.

Choosing ρ=C_2T/1-e^-T/16>0 (independent of k), (<ref>) implies that 
    (U^_k_L^2^2+A^_k_L^2^2)^1/2≤ρ
 if (U^0_L^2^2+A^0_L^2^2)^1/2≤ρ. Define 𝒯:B_ρ^2k→ B_ρ^2k by 𝒯(μ^_k(0),^_k(0))=(μ^_k(T),^_k(T)), where B_ρ^2k is the closed ball in ^2k of radius ρ and centered at the origin. Note that the map 𝒯 is continuous by the continuous dependence on initial conditions of the solution of ODEs. Thus, it has a fixed point by the Brouwer fixed point theorem, i.e., there exist (μ^_k(0),^_k(0))∈ B_ρ^2k such that (μ^_k(0),^_k(0))=(μ^_k(T),^_k(T)). Let U^0=∑_i=1^kμ^_ki(0)h_i and A^0=∑_i=1^k^_ki(0)h_i. Then U^0,A^0∈(h_1,⋯,h_k) and U^0=U^_k(T),A^0=A^_k(T).

With the choice of U^0 and A^0 we have (U^_k(s)_L^2^2+A^_k(s)_L^2^2)^1/2≤ρ for all s∈[0,T]. Hence

    (U^_k_L^∞(0,T;L^2(^3))^2+A^_k_L^∞(0,T;L^2(^3))^2)^1/2≤ρ.

Moreover, by integrating (<ref>) in s∈[0,T] and using U^_k(0)=U^_k(T), A^_k(0)=A^_k(T), we get 

    1/16∫_0^T(U^_k(s)_H^1^2+A^_k(s)_H^1^2)ds≤ C_2T.

Therefore, 

    U^_k_L^∞(0,T;L^2(^3))+A^_k_L^∞(0,T;L^2(^3))+U^_k_L^2(0,T;H^1(^3))+A^_k_L^2(0,T;H^1(^3))≤ C,

where C=√(4(ρ^2+16C_2T)) is independent of both  and k.

Using the uniform bounded sequences {U^_k}_k∈ and {A^_k}_k∈, and a standard limiting process, we get, for all >0, two T-periodic vector fields U^,A^∈ L^2(0,T;H^1_0(^3)) (both have -independent L^∞ L^2 and L^2H^1 bounds), a subsequence of {U^_k}_k∈, and a subsequence of {A^_k}_k∈ (still denoted by U^_k and A^_k, respectively) so that

   U^_k⇀U^, A^_k⇀A^  weakly in L^2(0,T;X),

   U^_k→U^, A^_k→A^  strongly in L^2(0,T;L^2(K)) for all compact sets K⊂^3,

   U^_k(s)⇀U^(s), A^_k(s)⇀A^(s)  weakly in L^2 for all s∈[0,T].

The weak convergence guarantees that U^(0)=U^(T) and A^(0)=A^(T). Moreover, the pair (U^,A^) is a periodic weak solution of the mollified perturbed Leray system (<ref>).

 

Step 2: A priori estimate of the pressure in the mollified perturbed Leray system

Note that ·ℒV=0 if · V=0. Therefore, by taking the divergence of (<ref>)_1, we obtain

-p^=   ∑_i,j=1^k_i_j[(η_*U^_i)U^_j+W_iU^_j+U^_i W_j+W_iW_j.

                .-(η_*A^_i)A^_j-D_iA^_j-A^_i D_j-D_iD_j].

Let 

p̃^=   ∑_i,j=1^kR_iR_j[(η_*U^_i)U^_j+W_iU^_j+U^_i W_j+W_iW_j.
                  .-(η_*A^_i)A^_j+D_iA^_j+A^_i D_j+D_iD_j],

where R_i denote the Riesz transforms. Note that p̃^ also satisfies (<ref>). We will show that p^=p̃^ up to an additive constant by proving (p^-p̃^)=0. 

Let V^(x,t)=(2t)^-1/2U^(y,s), π^(x,t)=(2t)^-1p^(y,s) and ℱ^(x,t)=(ℱ_1+ℱ^_2)(x,t) where 

    ℱ_1(x,t):=-1/(2t)^3/2(ℒW)(y,s),


ℱ^_2(x,t):=-1/(2t)^3/2   [W·U^+U^·W+(η_*U^)·U^+W·W.
   . -D·A^-A^·D-(η_*A^)·A^-D·D](y,s),

and y=x/√(2t) and s=log(√(2t)). Hence, ℱ^∈ L^∞(1,^2;H^-1(^3)), (V^,π) solves the non-stationary Stokes system on ^3×[1,^2] with force ℱ^ defined by (<ref>)_1, and V^ is in the energy class. According to the uniqueness of the solution to the forced, non-stationary Stokes system on ^3×[1,^2], we can conclude that π^=π̃^ where π̃^=(2t)^-1p̃^. Therefore (p^-p̃^)=0.

At this stage, we may replace p^ by p̃^. Recall that the Riesz transforms R_iϕ(x)=lim_→0^+∫_|x-y|>K_i(x-y)ϕ are Calderón-Zygmund operators since K_i(x)=x_i/|x|^n+1 are Calderón-Zygmund kernels. Applying the Calderón-Zygmund theory, we get

p^(s)_L^5/3≤    C[(η_*U^_i)U^_j+W_iU^_j+U^_i W_j+W_iW_j..
        ..-(η_*A^_i)A^_j+D_iA^_j+A^_i D_j+D_iD_j](s)_L^5/3
≤    C(U^(s)_L^10/3^2+A^(s)_L^10/3^2+W(s)_L^10/3^2+D(s)_L^10/3^2).

Hence we obtain the following a priori bound for p^:

p^_L^5/3(^3×[0,T])≤    C(U^_L^10/3(^3×[0,T])^2+A^_L^10/3(^3×[0,T])^2.
        .+W_L^10/3(^3×[0,T])^2+D_L^10/3(^3×[0,T])^2).

Recall that the sequences {U^}_>0 and {A^}_>0 are both bounded in L^∞ L^2 and L^2H^1 norms. So 

U^_L^10/3(^3×[0,T])=U^_L_y^10/3_L_s^10/3≤    U^_L_y^2^2/5U^_L_y^6^3/5_L_s^10/3

≤    U^_L^∞(0,T;L^2(^3))^2/5U^_L_y^6^3/5_L_s^10/3

≤    U^_L^∞(0,T;L^2(^3))^2/5U^_L^2(0,T;L^6(^3))^3/5

≲    U^_L^∞(0,T;L^2(^3))^2/5U^_L^2(0,T;H^1(^3))^3/5

≤    C,

where C is some constant independent of . Similarly, we also obtain 

A^_L^10/3(^3×[0,T])≤C.

In addition, because we are applying lem_2.5 with q=10/3 and =1/4, we have W_L^∞(0,T;L^10/3(^3))≤1/4 and D_L^∞(0,T;L^10/3(^3))≤1/4. Thus, we have the esitmates

W_L^10/3(^3×[0,T])≤1/4 T^10/3    and    D_L^10/3(^3×[0,T])≤1/4 T^10/3.


Using the bounds (<ref>)-(<ref>), (<ref>) implies that {p^}_>0 is a bounded sequence in L^5/3(^3×[0,T]).

 

Step 3: Convergence to a suitable periodic weak solution to (<ref>)

Since the sequences {U^}_>0 and {A^}_>0 are both bounded in L^∞(0,T;L^2(^3))- and L^2(0,T;H^1(^3))- norms, there exist U,A∈ L^∞(0,T;L^2(^3))∩ L^2(0,T;H^1_0(^3)) and two sequences {U^_k}_k∈,{A^_k}_k∈ such that

   U^_k⇀U, A^_k⇀A    weakly in L^2(0,T;X),

   U^_k→U, A^_k→A    strongly in L^2(0,T;L^2(K)) for all compact sets K⊂^3,

   U^_k(s)⇀U(s), A^_k(s)⇀A(s)    weakly in L^2 for all s∈[0,T],

as _k→0.

On the other hand, since {p^_k}_k∈ is a bounded sequence in L^5/3(^3×[0,T]), we have that 

p^_k⇀p  weakly in L^5/3(^3×[0,T]),

for some p∈ L^5/3(^3×[0,T]). Let u=U+W and a=A+D. The above convergences are enough to ensure that the triple (u,a,p) solves (<ref>) in the sense of distributions.

It remains to check that (u,a,p) satisfies the local energy inequality (<ref>). Note that (u^_k,a^_k,p^_k), where u^_k=U^_k+W and a^_k=A^_k+D, satisfies

    1.2{[                                                 ℒu^_k+W· u^_k+(η__k*U^_k)· U^_k+U^_k· W;                                             -D· a^_k-(η__k*A^_k)· A^_k-A^_k· D+ p^_k=0,;                                                 ℒa^_k+W· a^_k+(η__k*U^_k)· A^_k+U^_k· D;                                                   -D· u^_k-(η__k*A^_k)· U^_k-A^_k· W=0. ].


Testing (<ref>)_1 and (<ref>)_2 with u^_kψ and a^_kψ, respectively, where 0≤ψ∈ C^∞_0(^4) and adding them together, we get

       ∫_^4(|u^_k|^2+|a^_k|^2/2+|u^_k|^2+|a^_k|^2)ψ dyds
   =∫_^4|u^_k|^2+|a^_k|^2/2(_sψ+ψ)dyds+∫_^4|u^_k|^2+|a^_k|^2/2 (W-y)·ψ dyds

      +∫_^4(|U^_k|^2+2(U^_k·W)+|A^_k|^2+2(A^_k·D)/2 (η__k*U^_k).

                                                                      .+|W|^2+|D|^2/2 U^_k)·ψ dyds

      +∫_^4p^_k u^_k·ψ dyds

      -∫_^4((u^_k·a^_k)D+(U^_k·A^_k)(η__k*A^_k)+(U^_k·D)A^_k+(W·A^_k)A^_k.

                                                                             .+(W·D)A^_k)·ψ dyds

      +∫_^4((η__k*U^_k)-U^_k)·(W·U^_k+D·A^_k)ψ dyds

      +∫_^4((η__k*A^_k)-A^_k)·(U^_k·D+A^_k·W)ψ dyds.

Let 𝒦 be a compact subset of ^4. We have 

(η__k*U^_k)-U_L^2(𝒦)≤    (η__k*U^_k)-(η__k *U)_L^2(𝒦)+(η__k*U)-U_L^2(𝒦)
≤    U^_k-U_L^2(𝒦)+η__k*U-U_L^2(𝒦).

Since (η__k*U)(s)-U(s)_L_y^2≤(η__k*U)(s)_L_y^2+U(s)_L_y^2≤2U(s)_L_y^2∈ L_s^2(I) for all compact interval I, dominated convergence theorem implies that (η__k*U)-U_L^2(𝒦)→0 as _k→0. Together with the fact that U^_k→ U in L^2(0,T;L^2(K)) for all compact sets K⊂^3, we conclude that 

(η__k*U^_k)-U_L^2(𝒦)→0  as  _k→0    for all compact 𝒦⊂^4.
 
Similarly, we have

(η__k*A^_k)-A_L^2(𝒦)→0  as  _k→0    for all compact 𝒦⊂^4.

In addition, the sequence {u^}_>0 is bounded in L^10/3(^3×[0,T]) since it is bounded in L^2(0,T;L^6(^3)) and L^∞(0,T;L^2(^3)). According to the well-known fact mentioned in the Appendix of <cit.>, 

u^_k→u    strongly in L^5/2(𝒦)  as _k→0.


Combining (<ref>)-(<ref>) and the convergences in (<ref>) with the facts that W, D are locally differentiable and that the support of ψ is compact, each term on the right hand side of (<ref>) converges to the corresponding term involving u, U, a, A and p. On the other hand, ∫ |u^_k|^2dyds and ∫| a^_k|^2dyds are lower-semicontinuous as _k→0. This proves (<ref>) and completes the proof of thm_2.4_mhd.








 §.§ The time-periodic Leray system for the viscoelastic Navier-Stokes equations with damping

In this subsection, we follow the same approach as in Sect. 2.1 to construct a periodic weak solution to the Leray system for the viscoelastic Navier-Stokes equations with damping

    1.2.[ _su- u-u-y· u+(u·)u-n=13∑(g_n·)g_n+ p                                    =0; _sg_m- g_m-g_m-y· g_m+(u·)g_m-(g_m·)u                                    =0;                            · u =· g_m                                    =0 ]} in ^3×, m=1,2,3,
    lim_|y_0|→∞∫_B_1(y_0)(|u(y,s)-U_0(y,s)|^2+n=13∑|g_n(y,s)-G_n,0(y,s)|^2)dy=0  for all s∈,
    
    u(·,s)=u(·,s+T), g_m(·,s)=g_m(·,s+T)  in ^3 for all s∈, m=1,2,3,

for given T-periodic divergence-free vector fields U_0 and G_m,0, m=1,2,3. 

Periodic weak solutions and suitable periodic weak solutions of (<ref>) are defined as follows.
 Let U_0 and G_m,0, m=1,2,3, satisfy assum_2.1. A 4-tuple of vector fields (u,g_1,g_2,g_3) is a periodic weak solution to (<ref>) if for m=1,2,3 we have · u=· g_m=0, 
    u-U_0, g_m-G_m,0∈ L^∞(0,T;L^2(^3))∩ L^2(0,T;H^1(^3)),

and 

    ∫_0^T[(u,_sφ)-( u,φ)+(u+y· u-u· u+n=13∑(g_n·)g_n,φ)]ds=0,


    ∫_0^T[(g_m,_sφ)-( g_m,φ)+(g_m+y· g_m-u· g_m+g_m· u,φ)]ds=0,

holds for all φ∈𝒟_T.


 Let U_0 and G_m,0, m=1,2,3, satisfy assum_2.1.  
A 5-tuple (u,g_1,g_2, g_3,p) is a suitable periodic weak solution to (<ref>) if u,g_1,g_2,g_3,p are periodic in s with period T, (u,g_1,g_2,g_3) is a periodic weak solution to (<ref>), p∈ L^3/2_(^4), (u,g_1,g_2,g_3,p) solves (<ref>) in the sense of distributions, and the local energy inequality holds: 

∫_^4(|u|^2+|G|^2/2+|u|^2+|G|^2)ψ dyds≤   ∫_^4|u|^2+|G|^2/2(_sψ+ψ)dyds
   +∫_^4(|u|^2+|G|^2/2(u-y)+pu)·ψ dyds
   -n=13∑∫_^4(u·g_n)g_n·ψ dyds,
 where G=(g_1,g_2,g_3)∈^3×3, for all nonnegative ψ∈ C^∞_0(^4).


The main result of this subsection can be stated as the following:
thm_2.4_vNSEd
Assume U_0(y,s) and G_m,0(y,s), m=1,2,3, all satisfy assum_2.1 with q=10/3. Then (<ref>) has a periodic suitable weak solution (u,g_1,g_2,g_3,p) in ^4 with period T.


The proof follows from the same argument in that of thm_2.4_mhd. Let Z∈ C^∞(^3) with 0≤ Z≤ 1, Z(x)=1 for |x|>2 and Z(x)=0 for |x|<1. 
Applying lem_2.5 with =1/8, we are able to choose R_0=R_0(U_0,G_1,0,G_2,0,G_3,0)≥1 such that letting ξ(y)=Z(y/R_0) and setting

    W(y,s)=ξ(y)U_0(y,s)+w(y,s)

and 

    E_m(y,s)=ξ(y)G_m,0(y,s)+e_m(y,s), m=1,2,3,

where 
 
    w(y,s)=∫_^3_y 1/4π|y-z| _zξ(z)· U_0(z,s)dz

 and
  
    e_m(y,s)=∫_^3_y 1/4π|y-z| _zξ(z)· G_m,0(z,s)dz, m=1,2,3,

 W and E_m, m=1,2,3, all satisfy the conclusion of lem_2.5.

The Leray system (<ref>) can be written as 

    1.2{[ ℒu+(u·)u-n=13∑(g_n·)g_n+ p                         =0;       ℒg_m+(u·)g_m-(g_m·)u               =0, m=1,2,3,;                  · u=· g_m               =0, m=1,2,3, ].

where ℒ is given in (<ref>). We have to construct a solution of the form u=U+W and g_m=G_m+E_m, m=1,2,3. It follows that (U,G_1,G_2,G_3) satisfies the perturbed Leray system for the viscoelastic Navier-Stokes equations with damping

    1.2{[                                                                                                      ℒU+(W+U)· U+U· W;                                                            -n=13∑(E_n+G_n)· G_n-n=13∑G_n· E_n+ p=-ℛ_3(W,E_1,E_2,E_3),;                                                                                                ℒG_m+(W+U)· G_m+U· E_m;                                                                                     -(E_m+G_m)· U-G_m· W=-ℛ_4(W,E_m),;                                                                                                          · U=· G_m=0, ].

for m=1,2,3, where 

1.2
{[           ℛ_3(W,E_1,E_2,E_3)        :=ℒW+W·W-n=13∑E_n·E_n;                   ℛ_4(W,E_m) :=ℒE_m+W·E_m-E_m·W, m=1,2,3. ].


We first solve the following mollified perturbed Leray system for the viscoelastic Navier-Stokes equations with damping for (U^,G_1^,G_2^,G_3^,p^) in ^3×[0,T]:

    1.2{[                                                                                                             ℒU^+(W+(η_*U^))· U^+U^· W;                                                                    -n=13∑(E_n+(η_*G_n^))· G_n^-n=13∑G_n^· E_n+ p=-ℛ_3(W,E_1,E_2,E_3),;                                                                                                       ℒG_m^+(W+(η_*U^))· G_m^+U^· E_m;                                                                                             -(E_m+(η_*G_m^))· U^-G_m^· W=-ℛ_4(W,E_m),;                                                                                                                        · U^=· G_m^=0, ].

for m=1,2,3, where η_(y)=^-3η(y/) for some fixed function η∈ C^∞_0(^3) satisfying ∫_^3η dy=1. It has the following weak formulation:

    1.2{[                                     d/ds(U^,f)                                              =                         -( U^, f)+(U^+y· U^,f);                                                         -((η_*U^)· U^-n=13∑(η_*G_n^)· G_n^,f);                                                -(W· U^+U^· W-n=13∑E_n· G_n^-n=13∑G_n^· E_n,f);                                                                       -<ℛ_3(W,E_1,E_2,E_3),f>;                                   d/ds(G_m^,f)                                              =                   -( G_m^, f)+(G_m^+y· G_m^,f);                                                              -((η_*U^)· G_m^-(η_*G_m^)· U^,f);                                                          -(W· G_m^+U^· E_m-E_m· U^-G_m^· W,f);                                                                     -<ℛ_4(W,E_m),f>, m=1,2,3, ].

for all f∈𝒱 and a.e. s∈(0,T).

 

Step 1: Construction of a solution to the mollified perturbed Leray system


We use the Galerkin method to construct a solution of (<ref>). Let {h_k}_k∈⊂𝒱 be an orthonormal basis of H. For a fixed k∈, we look for an approximation solution of the form U^_k(y,s)=∑_i=1^kμ^_ki(s)h_i(y), (G^_m)_k(y,s)=∑_i=1^k(^_m)_ki(s)h_i(y), m=1,2,3. First, we prove the existence and derive an a priori bound for T-periodic solutions μ^_k=(μ^_k1,⋯,μ^_kk), (^_m)_k=((^_m)_k1,⋯,(^_m)_kk), m=1,2,3, to the system of ODEs

    1.2{[                                                                                                 d/dsμ^_kj =i=1k∑𝒜_ijμ^_ki+i=1k∑ n=13∑ℬ̃_ijn(^_n)_ki+i,l=1k∑𝒞^_iljμ^_kiμ^_kl-i,l=1k∑𝒞^_iljn=13∑(^_n)_ki(^_n)_kl+𝒟̃_j;                                                                                              d/ds(^_m)_kj                                     =i=1k∑ℰ̃_ijmμ^_ki+i=1k∑ℱ_ij(^_m)_ki+i,l=1k∑𝒢^_iljμ^_ki(^_m)_kl+ℋ̃_jm, ].

for j=1,⋯,k, where 𝒜_ij, 𝒞^_ilj, ℱ_ij and 𝒢^_ilj are the same as those in (<ref>), and 

ℬ̃_ijn   =(h_i·E_n,h_j)+(E_n·h_i,h_j),

𝒟̃_j   =-<ℛ_3(W,E_1,E_2,E_3),h_j>,

ℰ̃_ijm   =-(h_i·E_m,h_j)+(E_m·h_i,h_j),

ℋ̃_jm   =-<ℛ_4(W,E_m),h_j>.

Fix any k∈. For any U^0,G_m^0∈(h_1,⋯,h_k), m=1,2,3, there exist μ^_kj,(^_m)_kj∈ H^1(0,T̃), j=1,⋯,k, that uniquely solve (<ref>) with initial data μ^_kj(0)=(U^0,h_j), (^_m)_kj(0)=(G_m^0,h_j), j=1,⋯,k, for some 0<T̃≤ T. 

We prove that T̃=T. Indeed, multiplying the j-th equation of (<ref>)_1 by μ^_kj, multiplying the j-th equation of (<ref>)_2 by (^_m)_kj, and summing over all j=1,⋯,k and m=1,2,3, that yields

       1/2 d/ds(U^_k_L^2^2+n=13∑(G^_n)_k_L^2^2)+1/2(U^_k_L^2^2+n=13∑(G^_n)_k_L^2^2)

                                                                 +(U^_k_L^2^2+n=13∑(G^_n)_k_L^2^2)

   =-(U^_k·W-n=13∑(E_n·(G^_n)_k+(G^_n)_k·E_n),U^_k)

                                         -n=13∑((U^_k·E_n-E_n·U^_k)-(G^_n)_k·W,(G^_n)_k)

                                         -<ℛ_3(W,E_1,E_2,E_3),U^_k>-n=13∑<ℛ_4(W,E_n),(G^_n)_k>,

thanks to the vanishing of ((η_*U^)· U^,U^), (W· U^,U^), ((η_*U^)· G^_m,G^_m) and (W· G^_m,G^_m), and the cancellation of ∑_n=1^3((η_*G^_n)· G^_n,U^) and ((η_*G^_m)· U^,G^_m). Using lem_2.5 with =1/8, we get 

      |-(U^_k·W-n=13∑(E_n·(G^_n)_k+(G^_n)_k·E_n),U^_k).

       .-n=13∑((U^_k·E_n-E_n·U^_k)-(G^_n)_k·W,(G^_n)_k)|

   ≤1/16(7 U^_k_H^1^2+3 n=13∑(G^_n)_k_H^1^2),

and

      |-<ℛ_3(W,E_1,E_2,E_3),U^_k>-n=13∑<ℛ_4(W,E_n),(G^_n)_k>|

   ≤C_2+1/128(5 U^_k_H^1^2+3 n=13∑(G^_n)_k_H^1^2),

where C_2=32(ℒW_H^-1^2+∑_n=1^3ℒE_n_H^-1^2+(W_L^4^2+∑_n=1^3E_n_L^4^2)^2) is independent of s,T,k and ε.

Using the estimates (<ref>) and (<ref>), we obtain from (<ref>) the differential inequality

   d/ds(U^_k_L^2^2+n=13∑(G^_n)_k_L^2^2)+1/64(U^_k_L^2^2+n=13∑(G^_n)_k_L^2^2)

                                                      +1/64(U^_k_L^2^2+n=13∑(G^_n)_k_L^2^2)≤C_2.

The Gronwall inequality implies that 

e^s/64(U^_k_L^2^2+n=13∑(G^_n)_k_L^2^2)≤    (U^0_L^2^2+n=13∑G_n^0_L^2^2)+∫_0^T̃e^τ/64C_2dτ

≤    (U^0_L^2^2+n=13∑G_n^0_L^2^2)+e^T/64C_2T

for all s∈[0,T̃]. Since the right-hand side is finite, T̃ is not a blow-up time and we conclude that T̃=T.

Choosing ρ=C_2T/1-e^-T/64>0 (independent of k), (<ref>) implies that 
    (U^_k_L^2^2+∑_n=1^3(G^_n)_k_L^2^2)^1/2≤ρ
 if (U^0_L^2^2+n=13∑G_n^0_L^2^2)^1/2≤ρ. Define 𝒯:B_ρ^4k→ B_ρ^4k by 
    𝒯(μ^_k(0),(^_1)_k(0),(^_2)_k(0),(^_3)_k(0))=(μ^_k(T),(^_1)_k(T),(^_2)_k(T),(^_3)_k(T)),
 where B_ρ^4k is the closed ball in ^4k of radius ρ and centered at the origin. According to the continuous dependence on initial conditions of the solution of ODEs, the map 𝒯 is continuous. Thus, we can find a fixed point of 𝒯 by the Brouwer fixed point theorem. That is, there exist (μ^_k(0),(^_1)_k(0),(^_2)_k(0),(^_3)_k(0))∈ B_ρ^4k such that (μ^_k(0),(^_1)_k(0),(^_2)_k(0),(^_3)_k(0))=(μ^_k(T),(^_1)_k(T),(^_2)_k(T),(^_3)_k(T)). Let U^0=∑_i=1^kμ_ki(0)h_i and G_m^0=∑_i=1^k(_m)_ki(0)h_i. Then U^0,G_m^0∈(h_1,⋯,h_k) and U^0=U^_k(T),G_m^0=(G^_m)_k(T).

We have (U^_k(s)_L^2^2+n=13∑(G^_n)_k(s)_L^2^2)^1/2≤ρ for all s∈[0,T] by the choice of U^0 and G_m^0. Hence

    (U^_k_L^∞(0,T;L^2(^3))^2+∑_n=1^3(G^_n)_k_L^∞(0,T;L^2(^3))^2)^1/2≤ρ.

Moreover, by integrating (<ref>) in s∈[0,T] and using U^_k(0)=U^_k(T), (G^_m)_k(0)=(G^_m)_k(T), we get 

    1/64∫_0^T(U^_k(s)_H^1^2+∑_n=1^3(G^_n)_k(s)_H^1^2)ds≤ C_2T.

Therefore, 

   U^_k_L^∞(0,T;L^2(^3))+∑_n=1^3(G^_n)_k_L^∞(0,T;L^2(^3))
                                      +U^_k_L^2(0,T;H^1(^3))+∑_n=1^3+(G^_n)_k_L^2(0,T;H^1(^3))≤C,

where C=√(8(ρ^2+64C_2T)) is independent of both  and k.

Since the sequences {U^_k}_k∈ and {(G^_m)_k}_k∈ are uniformly bounded, a standard limiting process shows that, for all >0, we have, up to some subsequences, that 

   U^_k⇀U^, (G^_m)_k⇀G^_m  weakly in L^2(0,T;X),

   U^_k→U^, (G^_m)_k→G^_m  strongly in L^2(0,T;L^2(K)) for all compact sets K⊂^3,

   U^_k(s)⇀U^(s), (G^_m)_k(s)⇀G^_m(s)  weakly in L^2 for all s∈[0,T]

as k→∞, for some U^,(G^_m)∈ L^2(0,T;H^1_0(^3)), m=1,2,3, (all have -independent L^∞ L^2 and L^2H^1 bounds). The weak convergence ensures that U^(0)=U^(T) and (G^_m)(0)=G^_m(T). Furthermore, the 4-tuple (U^,G^_1,G^_2,G^_3) is a periodic weak solution of the mollified perturbed Leray system (<ref>).

 

Step 2: A priori estimate of the pressure in the mollified perturbed Leray system

By taking the divergence of (<ref>)_1, we obtain

-p^=   ∑_i,j=1^k_i_j [ (η_*U^_i)U^_j+W_iU^_j+U^_i W_j+W_iW_j

                  .-∑_n=1^3((η_*(G^_n)_i)(G^_n)_j+(E_n)_i(G^_n)_j+(G^_n)_i(E_n)_j+(E_n)_i(E_n)_j)].

Let 

p̃^=   ∑_i,j=1^kR_iR_j [ (η_*U^_i)U^_j+W_iU^_j+U^_i W_j+W_iW_j

                   .-∑_n=1^3((η_*(G^_n)_i)(G^_n)_j+(E_n)_i(G^_n)_j+(G^_n)_i(E_n)_j+(E_n)_i(E_n)_j)],

where R_i denote the Riesz transforms. Note that p̃^ also satisfies (<ref>). We will prove (p^-p̃^)=0 so that p^=p̃^ up to an additive constant by proving . 

Let V^(x,t)=(2t)^-1/2U^(y,s), π^(x,t)=(2t)^-1p^(y,s) and ℱ̃^(x,t)=(ℱ̃_1+ℱ̃^_2)(x,t) where 

    ℱ̃_1(x,t):=-1/(2t)^3/2(ℒW)(y,s),


ℱ̃^_2(x,t):=-1/(2t)^3/2     [  W·U^+U^·W+(η_*U^)·U^+W·W

     .-∑_n=1^3(E_n·G^_n-G^_n·E_n-(η_*G^_n)·G^_n-E_n·E_n)](y,s),

and y=x/√(2t) and s=log(√(2t)). Hence, ℱ̃^∈ L^∞(1,^2;H^-1(^3)), (V^,π) solves the non-stationary Stokes system on ^3×[1,^2] with force ℱ̃^ by (<ref>)_1, and V^ is in the energy class. In view of the uniqueness of the solution to the forced, non-stationary Stokes system on ^3×[1,^2], we can conclude that π^=π̃^ where π̃^=(2t)^-1p̃^. Therefore (p^-p̃^)=0.

At this point, we may replace p^ by p̃^. As before, the Calderón-Zygmund theory gives 

p^(s)_L^5/3≤    C   [ (η_*U^_i)U^_j+W_iU^_j+U^_i W_j+W_iW_j

        -.∑_n=1^3((η_*(G^_n)_i)(G^_n)_j(E_n)_i(G^_n)_j+(G^_n)_i(E_n)_j+(E_n)_i(E_n)_j)](s)_L^5/3

≤    C(U^(s)_L^10/3^2+∑_n=1^3G^_n(s)_L^10/3^2+W(s)_L^10/3^2+∑_n=1^3E_n(s)_L^10/3^2).

So we get the following a priori bound for p^:

p^_L^5/3(^3×[0,T])≤    C(U^_L^10/3(^3×[0,T])^2+∑_n=1^3G^_n_L^10/3(^3×[0,T])^2.

         +.W_L^10/3(^3×[0,T])^2+∑_n=1^3E_n_L^10/3(^3×[0,T])^2).

Since the sequences {U^}_>0 and {G^_m}_>0, m=1,2,3, are both bounded in L^∞ L^2 and L^2H^1 norms,

U^_L^10/3(^3×[0,T])=U^_L_y^10/3_L_s^10/3≤    U^_L_y^2^2/5U^_L_y^6^3/5_L_s^10/3

≤    U^_L^∞(0,T;L^2(^3))^2/5U^_L_y^6^3/5_L_s^10/3

≤    U^_L^∞(0,T;L^2(^3))^2/5U^_L^2(0,T;L^6(^3))^3/5

≲    U^_L^∞(0,T;L^2(^3))^2/5U^_L^2(0,T;H^1(^3))^3/5

≤    C,

where C is some constant independent of . Similarly, we have 

G^_m_L^10/3(^3×[0,T])≤C, m=1,2,3.

Moreover, since we are applying lem_2.5 with q=10/3 and =1/8, W_L^∞(0,T;L^10/3(^3))≤1/8 and E_m_L^∞(0,T;L^10/3(^3))≤1/8. Thus, we have the estimates

W_L^10/3(^3×[0,T])≤1/8 T^10/3    and    E_m_L^10/3(^3×[0,T])≤1/8 T^10/3, m=1,2,3.


Using the bounds (<ref>)-(<ref>), (<ref>) implies that {p^}_>0 is a bounded sequence in L^5/3(^3×[0,T]).

 

Step 3: Convergence to a suitable periodic weak solution to (<ref>)

On one hand, since the sequences {U^}_>0 and {G^_m}_>0, m=1,2,3, are all bounded in L^∞(0,T;L^2(^3))- and L^2(0,T;H^1(^3))- norms, there exist U,G_m∈ L^∞(0,T;L^2(^3))∩ L^2(0,T;H^1_0(^3)), m=1,2,3, and sequences {U^_k}_k∈,{G^_k_m}_k∈, m=1,2,3, such that for m=1,2,3,

   U^_k⇀U, G^_k_m⇀G_m    weakly in L^2(0,T;X),

   U^_k→U, G^_k_m→G_m    strongly in L^2(0,T;L^2(K)) for all compact sets K⊂^3,

   U^_k(s)⇀U(s), G^_k_m(s)⇀G_m(s)    weakly in L^2 for all s∈[0,T],

as _k→0.

On the other hand, since {p^_k}_k∈ is a bounded sequence in L^5/3(^3×[0,T]), we have that 

p^_k⇀p  weakly in L^5/3(^3×[0,T]),

for some p∈ L^5/3(^3×[0,T]). Let u=U+W and g_m=G_m+E_m, m=1,2,3. The above convergences are strong enough to guarantee that the 5-tuple (u,g_1,g_2,g_3,p) solves (<ref>) in the sense of distributions.

What is left is to show that (u,g_1,g_2,g_3,p) satisfies the local energy inequality (<ref>). Note that (u^_k,g^_k_1,g^_k_2,g^_k_3,p^_k), where u^_k=U^_k+W and g^_k_m=G^_k_m+E_m, m=1,2,3, satisfies

    1.2{[                                                                                                     ℒu^_k+W· u^_k+(η__k*U^_k)· U^_k+U^_k· W;                                                                       -n=13∑E_n· g^_k_n-n=13∑(η__k*G^_k_n)· G^_k_n-n=13∑G^_k_n· E_n+ p^_k=0;                                                                                             ℒg^_k_m+W· g^_k_m+(η__k*U^_k)· G^_k_m+U^_k· E_m;                                                                                                 -E_m· u^_k-(η__k*G^_k_m)· U^_k-G^_k_m· W=0. ].

Testing (<ref>)_1 and (<ref>)_2 for m=1,2,3, with u^_kψ and g^_k_mψ, m=1,2,3, respectively, where 0≤ψ∈ C^∞_0(^4) and adding them together, we get

       ∫_^4[1/2(|u^_k|^2+n=13∑ |g^_k_n|^2)+|u^_k|^2+n=13∑ |g^_k_n|^2]ψ dyds

   =∫_^41/2(|u^_k|^2+n=13∑ |g^_k_n|^2)(_sψ+ψ)dyds

      +∫_^41/2(|u^_k|^2+n=13∑ |g^_k_n|^2)(W-y)·ψ dyds

      +∫_^4[1/2(|U^_k|^2+2(U^_k·W)+n=13∑ (|G_n^_k|^2+2(G_n^_k·E_n)))(η__k*U^_k).

                                                         .+1/2(|W|^2+n=13∑ |E_n|^2)U^_k]·ψ dyds

      +∫_^4p^_k u^_k·ψ dyds

      -n=13∑ ∫_^4[(u^_k·g^_k_n)E_n+(U^_k·G^_k_n)(η__k*G^_k_n)+(U^_k·E_n)G^_k_n.

                                                       .+(W·G^_k_n)G^_k_n+(W·E_n)G^_k_n]·ψ dyds

      +∫_^4((η__k*U^_k)-U^_k)·(W·U^_k+n=13∑ E_n·G^_k_n)ψ dyds

      +n=13∑∫_^4((η__k*G^_k_n)-G^_k_n)·(U^_k·E_n+G^_k_n·W)ψ dyds.

Let 𝒦 be a compact subset of ^4. Using the same argument deriving (<ref>) and (<ref>), we have 

(η__k*U^_k)-U_L^2(𝒦)→0  as  _k→0    for all compact 𝒦⊂^4,
 
and, for m=1,2,3,

(η__k*G^_k_m)-G_m_L^2(𝒦)→0  as  _k→0    for all compact 𝒦⊂^4.

In addition, the sequence {u^}_>0 is bounded in L^10/3(^3×[0,T]) since it is bounded in L^2(0,T;L^6(^3)) and L^∞(0,T;L^2(^3)). As before, we use the fact in the Appendix of <cit.> to show that

u^_k→u    strongly in L^5/2(𝒦)  as _k→0.


Combining (<ref>)-(<ref>) and the convergences in (<ref>) with the facts that W, E_m are locally differentiable and that ψ is compactly supported, each term on the right hand side of (<ref>) converges to the corresponding term involving u, U, g_m, G_m and p. Passing limit as _k→0, we get the desired local energy inequality (<ref>) since ∫ |u^_k|^2dyds and ∫| g^_k_m|^2dyds are lower-semicontinuous as _k→0. This proves thm_2.4_vNSEd.









§ DISCRETELY SELF-SIMILAR SOLUTIONS

In this section, we prove thm_1.2_mhd and thm_1.2_vNSEd.


 §.§ Discretely self-similar solutions to the MHD equations


Let U_0(y,s)=√(2t)(e^tv_0)(x) and A_0=√(2t)(e^tb_0)(x). By lem_3.4, U_0 and A_0 both satisfy assum_2.1 with T=log and q=10/3. Let (u,a,p) be the T-periodic weak solution derived in thm_2.4_mhd. Let v(x,t)=u(y,s)/√(2t), b(x,t)=a(y,s)/√(2t) and π(x,t)=p(y,s)/2t where x,t,y,s satisfy (<ref>). Then (v,b,π) is a distributional solution to (<ref>).

Note that u-U_0 is periodic in s with period T=log(). So

v-e^t v_0_L_t^∞(1,^2;L_x^2(^3))^2≤    u-U_0_L_s^∞(1/2log2,1/2log2+log();L_y^2(^3))^2

≤    u-U_0_L_s^∞(0,T;L_y^2(^3))^2.

Similarly, we have

    v-e^t v_0_L_t^2(1,^2;L_x^2(^3))^2≤^3u-U_0_L_s^2(0,T;L_y^2(^3))^2,

and 

    _x(v-e^t v_0)_L_t^2(1,^2;L_x^2(^3))^2≤_y(u-U_0)_L_s^2(0,T;L_y^2(^3))^2.

Therefore,

v-e^t v_0∈L^∞(1,^2;L^2(^3))∩L^2(1,^2;H^1(^3)).



Note that v-e^t v_0 is -DSS because u-U_0 is T-periodic, where T=log(). For t>0, ^-2k≤ t< ^-2k+2 for some k∈ so 1≤^2kt<^2. Thus

v(t)-e^tv_0_L^2(^3)^2=    ^-1∫_^3|(v-e^tv_0)(x,^2t)|^2dx

=    ⋯

=    ^-k∫_^3|(v-e^tv_0)(x,^2kt)|^2dx

≤    t^1/2sup_1≤τ≤^2v(τ)-e^τv_0_L^2(^3)^2.

Moreover, 

∫_^-2k^^-2k+2∫_^3|(v(t)-e^tv_0)|^2dxdt=    ^-1∫_^-2k+2^^-2k+4∫_^3|(v(t)-e^tv_0)|^2dxdt

=    ⋯

=    ^-k∫_1^^2∫_^3|(v(t)-e^tv_0)|^2dxdt

implies that

∫_0^^2∫_^3|(v(t)-e^tv_0)|^2dxdt=    ∑_k=0^∞∫_^-2k^^-2k+2∫_^3|(v(t)-e^tv_0)|^2dxdt

=    (∑_k=0^∞^-k)∫_1^^2∫_^3|(v(t)-e^tv_0)|^2dxdt.

Therefore, we see from (<ref>) and (<ref>) that 

v-e^tv_0∈L^∞(0,^2;L^2(^3))∩L^2(0,^2;H^1(^3)).


We first prove that v has locally finite energy and enstrophy. In view of Remark 3.2 in <cit.>, we have

      sup_x_0∈^3∫_B_R(x_0)|e^tv_0(x)|^2dx
   =sup_x_0∈^3∫_B_1(x_0)|e^tv_0(R(x̃-x_0)+x_0)|^2R^3 dx̃

   =sup_x_0∈^3∫_B_1(x_0)|e^t/R^2v_0(x̃)|^2R^3 dx̃,  where v_0(x̃)=v_0(R(x̃-x_0)+x_0),

   =R^3 e^t/R^2v_0_L^2_^2

   ≲R^3 v_0_L^2_^2   (by Remark 3.2 in <cit.>)

   =sup_x_0∈^3∫_B_R(x_0)|v_0(x)|^2dx

   =^ksup_x_0∈^3∫_B_^-kR(^-kx_0)|v_0(x)|^2dx,  where ^k-1≤ R<^k for some k, (since v_0 is -DSS)

   ≤^ksup_x_0∈^3∫_B_1(^-kx_0)|v_0(x)|^2dx

   =^kv_0_L^2_^2.

Combining this result with (<ref>), we actually have

       0≤t<R^2 sup_x_0∈^3∫_B_R(x_0)|v(x,t)|^2dx

   ≤2(0≤t<R^2 sup_x_0∈^3∫_B_R(x_0)|v(x,t)-e^tv_0|^2dx+0≤t<R^2 sup_x_0∈^3∫_B_R(x_0)|e^tv_0|^2dx)

   ≲2(Rsup_1≤τ≤^2v(τ)-e^τv_0_L^2(^3)+Rv_0_^2)<∞.

Likewise, since 

sup_x_0∈^3∫_B_R(x_0)|_x(e^tv_0(x))|^2dxdt=    sup_x_0∈^3∫_B_1(x_0)|R^-1_x̃(e^tv_0)(R(x̃-x_0)+x_0)|^2R^3 dx̃

=    sup_x_0∈^3∫_B_1(x_0)|_x̃(e^t/R^2v_0)(x̃)|^2R dx̃

≲    R/(t/R^2)^1/2 v_0_L^2_^2   (by Remark 3.2 in <cit.>)

≤    /t^1/2 v_0_L^2_^2,

it follows from (<ref>) that

       sup_x_0∈^3∫_0^R^2∫_B_R(x_0)|v(x,t)|^2dxdt

   ≤2(sup_x_0∈^3∫_0^R^2∫_B_R(x_0)|(v-e^tv_0)|^2dxdt+sup_x_0∈^3∫_0^R^2∫_B_R(x_0)|(e^tv_0)|^2dxdt)

   ≤2(∑_m=0^∞^-(k+m)∫_1^^2∫_^3|(v(t)-e^tv_0)|^2dxdt+2Rv_0_L^2_^2)

   ≤2(R/-1∫_1^^2∫_^3|(v(t)-e^tv_0)|^2dxdt+2Rv_0_L^2_^2)<∞,

where k is some integer so that ^k-1≤ R<^k. The same conclusion of (<ref>) and (<ref>) can be drawn for b(t)-e^tb_0. This proves (<ref>).

Secondly, we prove the convergence to initial data. Let K be a compact subset of ^3. We split v(t)-v_0_L^2_ into two parts: v(t)-e^tv_0_L^2_ and e^tv_0-v_0_L^2_. The first part is controlled by (<ref>) as 

v(t)-e^tv_0_L^2(K)≲t^1/4→0  as t→0^+.

For the second part, we use the fact that e^tv_0→ v_0 in L^2_-3/2 as t→ 0^+ mentioned in the Remark 2.3 of <cit.>. Moreover, we have the embeddings L^3_w⊂ M^2,1⊂ L^2_-3/2⊂ L^2_ (see <ref> Appendix). Hence e^tv_0→ v_0 in L^2_-3/2 as t→ 0^+ implies

e^tv_0→v_0  in L^2_  as t→0^+.

Therefore, combining (<ref>) and (<ref>), we have 
v→v_0  in L^2_  as t→0^+.

The same convergence (<ref>) is true for b. This establishes the convergence to initial data.

Next, we prove the decay at spatial infinity. Fix any R>0. We split v into two parts: v-e^tv_0 and e^tv_0. For the first part, v-e^tv_0∈ L^2(0,R^2;L^2(^3)) since 
    ∫_0^R^2∫_^3|(v-e^tv_0)(x,t)|^2dxdt≤∫_0^R^2t^1/21≤τ≤^2sup(v-e^τv_0)(x,τ)_L^2(^3)^2dt<∞
 by (<ref>). The dominated convergence theorem then implies 

    ∫_0^R^2∫_B_R(x_0)|(v-e^tv_0)(x,t)|^2dxdt=∫_0^R^2∫_^3|(v-e^tv_0)(x,t)|^21_B_R(x_0)(x) dxdt→0

as |x_0|→∞. For the second part, since v_0 is -DSS, e^tv_0 is also -DSS and U_0 is periodic in s with the period T=log(). So (<ref>) and (<ref>) also hold for e^tv_0. In the same manner above, we can show 
    ∫_0^R^2∫_B_R(x_0)|e^tv_0(x)|^2dxdt→0
 as |x_0|→∞. Since the same proof works for b, we can conclude that (<ref>) holds.

Finally, the local energy inequality (<ref>) for (<ref>) follows from the local energy inequality (<ref>) for (<ref>).
 





 §.§ Discretely self-similar solutions to the viscoelastic Navier-Stokes equations with damping


Let U_0(y,s)=√(2t)(e^tv_0)(x) and (G_m)_0=√(2t)(e^t(f_0)_m)(x), m=1,2,3, where (f_0)_m is the m-th column of F_0. By lem_3.4, U_0 and (G_m)_0, m=1,2,3, all satisfy assum_2.1 with T=log and q=10/3. Let (u,g_1,g_2,g_3,p) be the T-periodic weak solution derived in thm_2.4_vNSEd. Let v(x,t)=u(y,s)/√(2t),  F(x,t)= G(y,s)/√(2t) and π(x,t)=p(y,s)/2t where G=(g_1,g_2,g_3) and x,t,y,s satisfy (<ref>). We skip the rest of the proof as it is essentially the same as that in Sect. <ref>.








§ SELF-SIMILAR SOLUTIONS

In this section, we prove thm_1.3_mhd and thm_1.3_vNSEd.


 §.§ Self-similar solutions to the MHD equations


Let U_0 and A_0 be defined as in Sect. <ref>. Since v_0 and b_0 are (-1)-homogeneous, 
    U_0(y)=2^3/2(4π)^-3/2∫_^3e^-|y-z|^2/2v_0(z)dz and A_0(y)=2^3/2(4π)^-3/2∫_^3e^-|y-z|^2/2b_0(z)dz
 are independent of s. By lem_3.4, U_0 and A_0 both satisfy assum_2.1 for any q∈(3,∞] because v_0 and b_0 are -DSS for all >1. Let W and D be defined as in (<ref>) and (<ref>), respectively. Then W and D are independent of s. Furthermore, according to lem_2.5, W and D satisfy the estimates (<ref>)-(<ref>) with q∈(3,∞]. Our goal is to solve the following variational form of the stationary Leray system for the MHD equations

1.2
{[ -(u,f)+(u+y·u-u·u+a·a,f)                       =0; -(a,f)+(a+y·a-u·a+a·u,f)                      =0, ]
.

for all f∈𝒱. Similar to the proof of thm_2.4_mhd, we are looking for a solution of the form u=W+U and a=D+A and using Galerkin method to achieve this. Note that (U,A) satisfies the perturbed stationary Leray system for the MHD equations, which has the weak formulation as 

1.2
{[                                          -(U,f)+(U+y·U,f)-(U·U-A·A,f);                                     =(W·U+U·W-D·A-A·D,f)+<ℛ_1(W,D),f>;                                          -(A,f)+(A+y·A,f)-(U·A-A·U,f);                                    =(W·A+U·D-D·U-A·W,f)+<ℛ_2(W,D),f>, ]
.

for all f∈𝒱, where ℛ_1 and ℛ_2 are the same as in (<ref>). Let {h_k}_k∈⊂𝒱 be an orthonormal basis of H. For a fixed k, we look for an approximation solution of the form U_k(y)=∑_i=1^kμ_kih_i(y), A_k(y)∑_i=1^k_kih_i(y). Plugging them into the weak formulation, we get the following algebraic system: 

    1.2{[ i=1k∑𝒜_ijμ_ki+i=1k∑ℬ_ij_ki+i,l=1k∑𝒞_iljμ_kiμ_kl-i,l=1k∑𝒞_ilj_ki_kl+𝒟_j                                                                     =0;                     i=1k∑ℰ_ijμ_ki+i=1k∑ℱ_ij_ki+i,l=1k∑𝒢_iljμ_ki_kl+ℋ_j                                                                    =0, ].

for j=1,⋯,k, where 𝒜_ij, ℬ_ij, 𝒟_j, ℰ_ij, ℱ_ij, ℋ_j are the same as those in (<ref>), and 

𝒞_ilj   =-(h_i·h_l,h_j),

𝒢_ilj   =-(h_i·h_l,h_j)+(h_l·h_i,h_j).

Let P:^2k→^2k be defined by

   (P(μ_k1,⋯,μ_kk,_k1,⋯,_kk))_j

       ={[ i=1k∑𝒜_ijμ_ki+i=1k∑ℬ_ij_ki+i,l=1k∑𝒞_iljμ_kiμ_kl-i,l=1k∑𝒞_ilj_ki_kl+𝒟_j,                                                                j=1,⋯,k,;       i=1k∑ℰ_i(j-k)μ_ki+i=1k∑ℱ_i(j-k)_ki+i,l=1k∑𝒢_il(j-k)μ_ki_kl+ℋ_j-k,                                                             j=k+1,⋯,2k. ].

From similar estimates as in (<ref>) and (<ref>), we have  that

   P(μ_k1,⋯,μ_kk,_k1,⋯,_kk)·(μ_k1,⋯,μ_kk,_k1,⋯,_kk)

=    -1/2(U_k_L^2^2+A_k_L^2^2)-(U_k_L^2^2+A_k_L^2^2)

    -(U_k·W-D·A_k-A_k·D,U_k)-(U_k·D-D·U_k-A_k·W,A_k)

    -<ℛ_1(W,D),U_k>-<ℛ_2(W,D),A_k>

≤    -1/2(U_k_L^2^2+A_k_L^2^2)-(U_k_L^2^2+A_k_L^2^2)+3/8(U_k_H^1^2+A_k_H^1^2)

    +C_2+3/32(U_k_H^1^2+A_k_H^1^2)

=    -1/32(U_k_L^2^2+A_k_L^2^2)-17/32(U_k_L^2^2+A_k_L^2^2)+C_2

≤    -1/32|(μ_k1,⋯,μ_kk,_k1,⋯,_kk)|^2+C_2

<    0,

if |(μ_k1,⋯,μ_kk,_k1,⋯,_kk)|=8√(C_2)=:ρ. Note that C_2 is independent of k. Thus, we obtain a point (μ_k1,⋯,μ_kk,_k1,⋯,_kk)∈ B_ρ^2k such that P(μ_k1,⋯,μ_kk,_k1,⋯,_kk)=0 by Brouwer's fixed point theorem. Then U_k(y)=∑_i=1^kμ_kih_i(y), A_k(y)∑_i=1^k_kih_i(y) is our approximation solution of (<ref>) with a priori bound 

    (U_k_L^2^2+A_k_L^2^2)+17( U_k_L^2^2+] A_k_L^2^2)≤32 C_2.

Therefore, we have, up to a subsequence, the following convergences

   U_k⇀U, A_k⇀A  weakly in H^1(^3),

   U_k→U, A_k→A  strongly in L^2(K) for all compact sets K⊂^3.

So we derive a solution (U,A) to (<ref>) with U,A∈ H^1(^3). Then (u,a), where u=U+W and a=A+D, is a solution to (<ref>). Note that u,a∈ H^1_∩ L^q for all 3<q≤ 6 since U,A∈ H^1⊂ L^q for q≤6 and W,D∈ L^q∩ L^4∩ C^∞_loc for q>3. 

Regarding the pressure, we define 
    p=∑_i,j=1^3R_iR_j(u_iu_j-a_ia_j),
 where R_i stands for the Riesz transforms. Then (u,a,p) satisfies the stationary Leray system for the MHD equations (<ref>) in the sense of distributions. Moreover, Calderon-Zygmund estimates gives the following a priori bound for p: for 3<q≤6 
    p_L^q/2(^3)≤ Cu_L^q(^3)^2.


Recovering (v,b,π) from (u,a,p) by the relation (<ref>), we obtain a self-similar weak solution of (<ref>) (see <cit.>). It remains to show that (v,b,π) is a local Leray solution of (<ref>).

Recall that (U,p) is a solution of the stationary Stokes system with the force 

    𝒢_1=U+y· U-(U· U-A· A)-W· U-U· W+D· A+A· D-ℒW-W· W+D· D.

Applying the regularity result in <cit.> on compact subsets of ^3, u and p are actually smooth. Additionally, A is a solution of the Poisson equation with the right hand side 

    𝒢_2=A+y· A-(U· A-A· U)-W· A-U· D+D· U+A· W-ℒD-W· D+D· W.

A standard elliptic regularity result leads to the smoothness for A on compact subsets of ^3. Thus, u,a and p inherit the smoothness from U,W,A and D. Therefore, from the self-similarity of v,b and π, they are smooth in both spatial and time variables. Consequently, the local energy inequality (<ref>) can be achieved via integrating by parts. The rest of conditions from def_loc_leray_mhd and the estimates of the distance between the solution (v,b) and the background (e^tv_0,e^tb_0) can be verified using the same approach as in Sect. <ref>.






 §.§ Self-similar solutions to the viscoelastic Navier-Stokes equations with damping


The proof is basically the same as in Sect. <ref>. It is worth noting that in (<ref>) we use the estimates (<ref>) and (<ref>) obtained by applying lem_2.5 with =1/4; while here we acheive (<ref>) from estimates (<ref>) and (<ref>) by applying the same lemma but with the parameter =1/8. The details of verification are left to the reader.










§ APPENDIX

In this appendix, we prove the three inclusions L^3_w⊂ M^2,1⊂ L^2_-3/2⊂ L^2_. To begin with, the first inclusion can be shown by the inequality

r^-1∫_B_r(x_0)|f(x)|^2dx=    r^-1∫_B_r(x_0)∫_0^|f(x)|2 ddx

=    r^-1∫_B_r(x_0)∫_0^∞21_|f|>(x) ddx

=    r^-1∫_0^∞2|{|f|>}∩B_r(x_0)|d

=    r^-1∫_0^r^-12|{|f|>}∩B_r(x_0)|d+r^-1∫_r^-1^∞2|{|f|>}∩B_r(x_0)|d

≤    r^-1∫_0^r^-12|B_r(x_0)|d+r^-1∫_r^-1^∞2|{|f|>}|d

≤    r^-1|B_r(x_0)|r^-2+r^-1∫_r^-1^∞2f_L^3_w^3^-3d

≲    1+f_L^3_w^3.
 
Next, the second inclusion is valid as

∫_^3|f(x)|^2/(1+|x|)^3 dx=    ∫_|x|<1|f(x)|^2/(1+|x|)^3 dx+∑_k=0^∞∫_2^k≤|x|<2^k+1|f(x)|^2/(1+|x|)^3 dx

≤    ∫_B_1(0)|f(x)|^2dx+∑_k=0^∞1/(1+2^k)^3∫_2^k≤|x|<2^k+1|f(x)|^2dx

≤    f_M^2,1^2+∑_k=0^∞1/(1+2^k)^3 2^k+1f_M^2,1^2

≲    f_M^2,1^2.

Finally, the third inclusion holds since

∫_|x|≤M|f(x)|^2dx≤(1+M)^3∫_^3|f(x)|^2/(1+|x|)^3 dx=(1+M)^3f_L^2_-3/2^2.






§ ACKNOWLEDGMENTS


The research was partially supported by FYF (#6456) of Graduate and Postdoctoral Studies, University of British Columbia (BC). The author would like to express his fully gratitude to Tai-Peng Tsai for kindly discussion. Also, he thanks Anyi Bao for her proofreading.

 
 
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