Liouville-type theorems for the Navier–Stokes equations

Liouville type of theorems with weights for the Navier-Stokes equations and the Euler equations

We study Liouville type of theorems for the Navier-Stokes and the Euler equations on R N , N ≥ 2. Specifically, we prove that if a weak solution (v, p) satisfies |v| 2 +|p| ∈ L 1 (0, T ; L 1 (R N , w 1 (x)dx)) and R N p(x, t)w 2 (x)dx ≥ 0 for some weight functions w 1 (x) and w 2 (x), then the solution is trivial, namely v = 0 almost everywhere on R N × (0, T). Similar results hold for the MHD ...

Liouville-Type Theorems for Some Integral Systems

/2 ( ) ( ) u u        where  is the Fourier transformation and  its inverse. The question is to determine for which values of the exponents pi and qi the only nonnegative solution (u, v) of (1) and (2) is trivial, i.e., (u; v) = (0, 0). When 2   , is the case of the Emden-Fowler equation 0 , 0     u u u k in N  (5) When ) 3 )( 2 /( ) 2 ( 1      N N N k , it has been prove...

Liouville theorems on some indefinite equations

In this note, we present some Liouville type theorems about the nonnegative solutions to some indefinite elliptic equations.

Liouville type theorems and Harnack type inequalities for semilinear elliptic equations

Liouville type of theorems for the Euler and the Navier-Stokes equations

We prove Liouville type of theorems for weak solutions of the Navier-Stokes and the Euler equations. In particular, if the pressure satisfies p ∈ L1(0, T ;H1(RN )), then the corresponding velocity should be trivial, namely v = 0 on RN × (0, T ), while if p ∈ L1(0, T ;L1(RN )), then we have equipartition of energy over each component. Similar results hold also for the magnetohydrodynamic equations.