Local well-posedness for a moving rigid region in Surface Quasi-Geostrophic equations
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arXiv
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2026
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| author | Godard-Cadillac, Ludovic Roy, Arnab |
| author_facet | Godard-Cadillac, Ludovic Roy, Arnab |
| contents | We introduce and analyze a class of Surface Quasi-Geostrophic (SQG) equations in the presence of moving rigid obstacles. The model is motivated both by vortex-wave type asymptotics for singular structures in active scalar equations and by geophysical phenomena exhibiting rigid-like coherent regions, such as cyclone eyes or long-lived atmospheric dust clouds. We consider the critical SQG equation in a time-dependent exterior domain generated by a prescribed rigid motion and reconstruct the velocity through a nonlocal elliptic formulation adapted to impermeability constraints. The active scalar is assumed to remain constant inside the rigid region and in a neighborhood of its boundary, yielding a plateau structure compatible with the transport dynamics. For a single moving obstacle, we establish local well-posedness of classical solutions in Sobolev spaces $H^k$, $k\geq 4$ together with uniqueness, local stability, and a blow-up criterion. The analysis relies on a reformulation in adapted coordinates reducing the problem to a fixed domain, combined with integral representations for the fractional elliptic operator, regularization procedures, and a nonlinear fixed-point argument. A central difficulty comes from the critical singularity of the SQG Biot-Savart kernel in the case $s=\frac{1}{2}$, for which the velocity reconstruction near the moving boundary requires commutator estimates. We further prove propagation of the plateau property and derive a priori estimates controlling both the support of the scalar gradient and the Sobolev norm of the solution. This work provides, to our knowledge, the first well-posedness theory for SQG equations with moving rigid obstacles and constitutes a first step toward the rigorous derivation of point-vortex type dynamics from shrinking rigid bodies in SQG flows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23289 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local well-posedness for a moving rigid region in Surface Quasi-Geostrophic equations Godard-Cadillac, Ludovic Roy, Arnab Analysis of PDEs 35Q35, 76B03, 35A01 We introduce and analyze a class of Surface Quasi-Geostrophic (SQG) equations in the presence of moving rigid obstacles. The model is motivated both by vortex-wave type asymptotics for singular structures in active scalar equations and by geophysical phenomena exhibiting rigid-like coherent regions, such as cyclone eyes or long-lived atmospheric dust clouds. We consider the critical SQG equation in a time-dependent exterior domain generated by a prescribed rigid motion and reconstruct the velocity through a nonlocal elliptic formulation adapted to impermeability constraints. The active scalar is assumed to remain constant inside the rigid region and in a neighborhood of its boundary, yielding a plateau structure compatible with the transport dynamics. For a single moving obstacle, we establish local well-posedness of classical solutions in Sobolev spaces $H^k$, $k\geq 4$ together with uniqueness, local stability, and a blow-up criterion. The analysis relies on a reformulation in adapted coordinates reducing the problem to a fixed domain, combined with integral representations for the fractional elliptic operator, regularization procedures, and a nonlinear fixed-point argument. A central difficulty comes from the critical singularity of the SQG Biot-Savart kernel in the case $s=\frac{1}{2}$, for which the velocity reconstruction near the moving boundary requires commutator estimates. We further prove propagation of the plateau property and derive a priori estimates controlling both the support of the scalar gradient and the Sobolev norm of the solution. This work provides, to our knowledge, the first well-posedness theory for SQG equations with moving rigid obstacles and constitutes a first step toward the rigorous derivation of point-vortex type dynamics from shrinking rigid bodies in SQG flows. |
| title | Local well-posedness for a moving rigid region in Surface Quasi-Geostrophic equations |
| topic | Analysis of PDEs 35Q35, 76B03, 35A01 |
| url | https://arxiv.org/abs/2605.23289 |