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Ill posedness for the full Euler system driven by multiplicative white noise. (English) Zbl 1496.35294

The paper under review establishes the ill-posedness of weak solutions (more specifically, the existence of infinitely many global-in-time weak solutions) to the full Euler system describing the motion of a compressible fluid driven by a multiplicative white noise in dimension 2 or 3. In [D. Breit et al., Anal. PDE 13, No. 2, 371–402 (2020; Zbl 1435.35289)], it was shown that the isentropic Euler system driven by a general additive/multiplicative white noise is ill-posed, but the infinitely many weak solutions constructed therein may be physically irrelevant, in the sense that they may experience an initial energy jump. In contrast, the infinitely many weak solutions shown to exist in this paper are physical admissible – they conserve total energy and satisfy an entropy inequality.
Key ingredients of the proof include transforming the problem into PDE with random parameters, applying a result of C. De Lellis and L. Székelyhidi jun. [Arch. Ration. Mech. Anal. 195, No. 1, 225–260 (2010; Zbl 1192.35138)] for incompressible Euler with constant pressure obtained via convex integration, and employing a pasting argument for the particular solutions obtained as above. The strategy of the arguments follows essentially T. Luo et al. [Adv. Math. 291, 542–583 (2016; Zbl 1337.35114)].

MSC:

35Q31 Euler equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35D30 Weak solutions to PDEs
60H40 White noise theory
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35R60 PDEs with randomness, stochastic partial differential equations
35R25 Ill-posed problems for PDEs
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References:

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