The lectures that follow present new possibilities of defining generalized flows and generalized solutions to the Euler equations
; explore the existence, uniqueness, and long-time behavior of hydrodynamic evolution equations; and review recent progress in understanding small-scale and finite time singularity formation in solutions of the incompressible Euler equation
Consequently, by substituting equations (16), (17), (20), and (21) into equation (15), the linearized Euler equation
can be obtained as follows:
The full-order solver uses the cell-centered finite volume method to solve the Euler equation
. The AUSM+ scheme is used to discretize the computation domain , while the implicit LU-SGS scheme is used for temporal integration .
If we assume that radiation profiles s and the variable o, vary little, so we can write an expression very similar to the Euler equation
These two decisions are characterized by final-age version of the static labor supply Euler equation
(24) and the static intended bequests Euler equation
We first describe the variational formulation of the problem, including the Euler equation
governing the equilibrium drops that results from the first variation of the energy functional.
The economy is being held back not by exogenous "headwinds" but instead, since interest rates in the past (initially, in the second half of the 1990s) were too low and too much spending was brought forward from the future, is being held back by the endogenous working of the Euler equation
for consumption, which says that consumption will be on a downward path, relative to income, if the relevant interest rate is below the rate of time preference (and, heuristically, there is reason to believe that this is the case).
We consider the lax problem, which is the Euler equation
(61) and has the following starting conditions:
However, it gives us an important intertemporal optimality condition for consumption known as the Euler equation
. To derive it, we use the chain rule to express the time derivative of .J'([W.sub.t]) as
The first answer simply uses the law of iterated expectations, where we take an Euler equation
and project it on the forecasters' information set (actually, the forecasters do the projecting for us).
The aggregate demand is derived from the representative household's Euler equation
. After imposing equilibrium conditions, the log-linear form of the Euler equation
This is problematic, because GDP consists of consumption and investment, and what comes down most during an economic disaster is investment, not consumption (which enters the Euler equation
and thus matters for pricing.) During the Great Depression, for example, real GDP fell by 30 percent but consumption only dropped by 10 percent.