## On well-posedness and singularity formation for the Euler-Riesz system.(English)Zbl 1477.35143

Summary: In this paper, we investigate the initial value problem for the Euler-Riesz system, where the interaction forcing is given by $$\operatorname{\nabla} ( - \Delta )^s \rho$$ for some $$- 1 < s < 0$$, with $$s = - 1$$ corresponding to the classical Euler-Poisson system. We develop a functional framework to establish local-in-time existence and uniqueness of classical solutions for the Euler-Riesz system. In this framework, the fluid density could decay fast at infinity, and the Euler-Poisson system can be covered as a special case. Moreover, we prove local well-posedness for the pressureless Euler-Riesz system when the potential is repulsive, by observing hyperbolic nature of the system. Finally, we present sufficient conditions on the finite-time blowup of classical solutions for the isentropic/isothermal Euler-Riesz system with either attractive or repulsive interaction forces. The proof, which is based on estimates of several physical quantities, establishes finite-time blowup for a large class of initial data; in particular, it is not required that the density is of compact support.

### MSC:

 35Q31 Euler equations 35Q35 PDEs in connection with fluid mechanics 35B44 Blow-up in context of PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35A09 Classical solutions to PDEs
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