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On admissibility criteria for weak solutions of the Euler equations. (English) Zbl 1192.35138
The authors consider the Cauchy problem for the incompressible Euler equations in $$n$$ space dimensions, $$n\geq 2$$,
\begin{aligned} \frac{\partial v}{\partial t}+\text{div}\,(v\otimes\,v)+\nabla p=0, \quad \text{div}\,v=0 &\quad x\in\mathbb R^n,\;t>0,\\ v(x,0)=v_0(x), &\quad x\in\mathbb R^n, \end{aligned}\tag{$$*$$} where $$v_0$$ is a given divergence-free vector. The main result of the paper is the non-uniqueness theorem to the problem $$(*)$$. It is proved that there exist bounded and compactly supported $$v_0$$ for which there are
(1)
infinitely many weak solutions of $$(*)$$ satisfying both the strong and local energy equalities;
(2)
weak solutions of $$(*)$$ satisfying the strong energy inequality but not the energy equality;
(3)
weak solutions of $$(*)$$ satisfying the weak energy inequality but not the strong energy inequality.
Another non-uniqueness result is obtained to the system of isentropic gas dynamics in Eulerian coordinates
\begin{aligned} \frac{\partial \rho}{\partial t}+\text{div}(\rho v)=0, &\quad x\in\mathbb R^n,\;t>0,\\ \frac{\partial }{\partial t}(\rho v)+\text{div}(\rho v\otimes\,v) +\nabla [p(\rho)]=0, &\quad x\in\mathbb R^n,\;t>0,\\ v(x,0)=v_0(x),\quad \rho(x,0)=\rho_0(x), &\quad x\in\mathbb R^n. \end{aligned} Here $$v$$ is the velocity of a gas, $$\rho$$ is the density, the pressure $$p$$ is a function of $$\rho$$.
The proves are based on the Baire category method.

##### MSC:
 35Q31 Euler equations 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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