More Like this

1. Rough solutions of the relativistic Euler equations

   https://doi.org/10.1142/S0219891624500127  

   Yu, Sifan (June 2024, Journal of Hyperbolic Differential Equations)

   We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm [Formula: see text] of the variables in the “wave-part” and the Hölder norm [Formula: see text] of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption [Formula: see text] is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory. 

   more » « less
2. The Neumann function and the L ^p Neumann problem in chord-arc domains

   https://doi.org/10.1515/ans-2023-0171  

   Hofmann, Steve; Sparrius, Derek (February 2025, Advanced Nonlinear Studies)

   Abstract We construct the Neumann function in a 1-sided chord-arc domain (i.e., a uniform domain with an Ahlfors regular boundary), and establish size and Hölder continuity estimates up to the boundary. We then obtain a Kenig-Pipher type theorem, in whichL^psolvability of the Neumann problem is shown to yield solvability inL^qfor 1 <q<p, and in the Hardy spaceH¹, in 2-sided chord-arc domains, under suitable background hypotheses. 

   more » « less
3. The α$$\alpha$$‐SQG patch problem is illposed in C2,β and W2,p

   https://doi.org/10.1002/cpa.22236  

   Kiselev, Alexander; Luo, Xiaoyutao (April 2025, Communications on Pure and Applied Mathematics)

   Abstract We consider the patch problem for the ‐(surface quasi‐geostrophic) SQG system with the values and being the 2D Euler and the SQG equations respectively. It is well‐known that the Euler patches are globally wellposed in non‐endpoint Hölder spaces, as well as in , spaces. In stark contrast to the Euler case, we prove that for , the ‐SQG patch problem is strongly illposed ineveryHölder space with . Moreover, in a suitable range of regularity, the same strong illposedness holds foreverySobolev space unless . 

   more » « less
4. Regularity of almost-minimizers of Hölder-coefficient surface energies

   https://doi.org/10.3934/dcds.2022015  

   Simmons, David A (January 2022, Discrete and Continuous Dynamical Systems)

   We study almost-minimizers of anisotropic surface energies defined by a Hölder continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally Hölder continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument. 

   more » « less
5. Watershed‐Scale Effective Hydraulic Properties of the Continental United States

   https://doi.org/10.1029/2020MS002440  

   Tashie, Arik; Pavelsky, Tamlin; Band, Lawrence; Topp, Simon (June 2021, Journal of Advances in Modeling Earth Systems)

   Abstract In land surface models (LSMs), the hydraulic properties of the subsurface are commonly estimated according to the texture of soils at the Earth's surface. This approach ignores macropores, fracture flow, heterogeneity, and the effects of variable distribution of water in the subsurface oneffectivewatershed‐scale hydraulic variables. Using hydrograph recession analysis, we empirically constrain estimates of watershed‐scale effective hydraulic conductivities (K) and effective drainable aquifer storages (S) of all reference watersheds in the conterminous United States for which sufficient streamflow data are available (n = 1,561). Then, we use machine learning methods to model these properties across the entire conterminous United States. Model validation results in high confidence for estimates of log(K) (r² > 0.89; 1% < bias < 9%) and reasonable confidence forS(r² > 0.83; −70% < bias < −18%). Our estimates of effectiveKare, on average, two orders of magnitude higher than comparable soil‐texture‐based estimates of averageK, confirming the importance of soil structure and preferential flow pathways at the watershed scale. Our estimates of effectiveScompare favorably with recent global estimates of mobile groundwater and are spatially heterogeneous (5–3,355 mm). Because estimates ofSare much lower than the global maximums generally used in LSMs (e.g., 5,000 mm in Noah‐MP), they may serve both to limit model spin‐up time and to constrain model parameters to more realistic values. These results represent the first attempt to constrain estimates of watershed‐scale effective hydraulic variables that are necessary for the implementation of LSMs for the entire conterminous United States. 

   more » « less