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
infinitely many weak solutions of \((*)\) satisfying both the strong and local energy equalities;
weak solutions of \((*)\) satisfying the strong energy inequality but not the energy equality;
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.


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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