{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T10:27:37Z","timestamp":1786616857985,"version":"build-2736575974"},"reference-count":33,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T00:00:00Z","timestamp":1786579200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We develop a KvN\/Weyl\/Wigner\/Moyal phase-space formulation for shock-containing compressible, irrotational, barotropic Euler flow. Smooth branches are represented by branchwise Wigner distributions, while piecewise-smooth entropy-admissible shocks generate an interface-supported defect in the weak phase-space balance. This defect is concentrated on the moving shock surface and is weighted by the normal relative transport flux between the one-sided branches. An exact planar constant-state three-dimensional example shows how the same mass flux is transferred between distinct velocity-space supports and how its moments recover the classical jump structure. We also introduce a shock solution of the one-dimensional Burgers equation with a triangular initial profile as an exactly solvable reduced benchmark. In this example, the shock trajectory, transported branch weights, branchwise Wigner transforms, and a two-component localized phase-space defect are obtained in closed form. The construction is a restricted branchwise representation of Euler shocks already selected by the Rankine\u2013Hugoniot and entropy conditions; it is not a new admissibility criterion or a complete global Wigner theory across discontinuities. The formulation separates smooth phase-space evolution from singular interface contributions within a unified construction and provides a compact diagnostic description of shock-supported phase-space structure.<\/jats:p>","DOI":"10.3390\/e28080906","type":"journal-article","created":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T09:31:09Z","timestamp":1786613469000},"page":"906","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Phase-Space Formulation of Shock-Containing Irrotational Barotropic Euler Flow"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4227-956X","authenticated-orcid":false,"given":"Sandor M.","family":"Molnar","sequence":"first","affiliation":[{"name":"Institute of Astronomy and Astrophysics, Academia Sinica, No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-4210-2760","authenticated-orcid":false,"given":"Joseph R.","family":"Godfrey","sequence":"additional","affiliation":[{"name":"Grado Department of Industrial and Systems Engineering, Virginia Tech, Alexandria, VA 22305, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1103\/PhysRevLett.45.790","article-title":"Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics","volume":"45","author":"Morrison","year":"1980","journal-title":"Phys. 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