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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. 

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2. The stationary Boussinesq problem under singular forcing

   https://doi.org/10.1142/S0218202521500196  

   Allendes, Alejandro; Otárola, Enrique; Salgado, Abner J. (April 2021, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   In Lipschitz two- and three-dimensional domains, we study the existence for the so-called Boussinesq model of thermally driven convection under singular forcing. By singular we mean that the heat source is allowed to belong to [Formula: see text], where [Formula: see text] is a weight in the Muckenhoupt class [Formula: see text] that is regular near the boundary. We propose a finite element scheme and, under the assumption that the domain is convex and [Formula: see text], show its convergence. In the case that the thermal diffusion and viscosity are constants, we propose an a posteriori error estimator and show its reliability. We also explore efficiency estimates. 

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3. Stationary Cahn–Hilliard–Navier–Stokes equations for the diffuse interface model of compressible flows

   https://doi.org/10.1142/S0218202520500475  

   Liang, Zhilei; Wang, Dehua (November 2020, Mathematical Models and Methods in Applied Sciences)

   null (Ed.)

   A system of partial differential equations for a diffusion interface model is considered for the stationary motion of two macroscopically immiscible, viscous Newtonian fluids in a three-dimensional bounded domain. The governing equations consist of the stationary Navier–Stokes equations for compressible fluids and a stationary Cahn–Hilliard type equation for the mass concentration difference. Approximate solutions are constructed through a two-level approximation procedure, and the limit of the sequence of approximate solutions is obtained by a weak convergence method. New ideas and estimates are developed to establish the existence of weak solutions with a wide range of adiabatic exponent. 

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4. Fast Diffusion leads to partial mass concentration in Keller–Segel type stationary solutions

   https://doi.org/10.1142/S021820252250018X  

   Carrillo, J. A.; Delgadino, M. G.; Frank, R. L.; Lewin, M. (April 2022, Mathematical Models and Methods in Applied Sciences)

   We show that partial mass concentration can happen for stationary solutions of aggregation–diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions [Formula: see text]. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions [Formula: see text], for homogeneous interaction potentials with higher power. 

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5. A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs

   https://doi.org/10.3934/dcdss.2022012  

   Antil, Harbir; Gal, Ciprian G.; Warma, Mahamadi (January 2022, Discrete and Continuous Dynamical Systems - S)

   We consider optimal control of fractional in time (subdiffusive, i.e., for \begin{document}$$ 0<\gamma <1 $$\end{document}) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we \begin{document}$$\mathsf{first\;show}$$\end{document} the existence and regularity of solutions to the forward and the associated \begin{document}$$\mathsf{backward\;(adjoint)}$$\end{document} problems. In the second part, we prove existence of optimal \begin{document}$$\mathsf{controls }$$\end{document} and characterize the associated \begin{document}$$\mathsf{first\;order}$$\end{document} optimality conditions. Several examples involving fractional in time (and some fractional in space diffusion) equations are described in detail. The most challenging obstacle we overcome is the failure of the semigroup property for the semilinear problem in any scaling of (frequency-domain) Hilbert spaces. 

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