# Acknowledgement

The work of D.S. and V.V. was supported by the Russian Ministry of Education and Science (Grant No. 14.132.21.1687), the Russian Foundation for Basic Research (Grant No. 11-07-00464), and Russian Academy of Science. The work at RIEC was supported by the Japan Science and Technology Agency, CREST and by the Japan Society for Promotion of Science. The authors are grateful to S.O. Yurchenko for helpful discussions, and to Prof. M. Polini for valuable comments.

# Calculation of Average Values

The statistical average values can be exactly calculated with the distribution function (9.1) for linear energy spectrum. The particle density is given by

The last term could be evaluated as [1 - *(?] ~ ^{3/2},* while the remainder is nothing more but the particle density in the absence of flow

*n*Similarly, the energy density reads as follows:

_{0}.

The stress tensor is

The density of inverse energy *(r ^{1})* can be expressed in terms of elementary functions

The following relations for the derivatives of average values are required to represent the Euler equation in the canonical form:

# Weak Nonlocality Approximation for Poisson Equation

One solves the Poisson equation for the gated 2D electron gas (2DEG). The 2DEG plane is *z* = 0, the grounded top and bottom gates are placed *atz = d** _{1}* and

*z = -d*respectively, and the top and bottom dielectric permittivities are

_{ъ}*к*and k

_{г}_{2}. We assume that electron density in 2DEG varies only in the

*x*direction and write the Poisson equation as

with boundary conditions

Here *o= -en* is the 2D charge density, and *n* is the electron density.

f+°°

After the Fourier transform *tp _{k}* =

*tp{x,z)e dx*the Poisson equation becomes

Solving Eq. (9.B5) with boundary conditions (9.B2)-(9.B4), which apply to the Fourier components as well, we obtain

Assuming that the 2D electron density varies slowly (/cdj * _{2}* -С 1), we expand Eq. (9.B6) in series over /(and arrive at the final solution after inverse Fourier transform:

Equation (9.B7) is further simplified to Eq. (9.22) for equal permittivities of top- and bottom gate dielectrics *к _{г} = к*

_{2}*= к.*

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