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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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