arXiv subjects

Jingchuang Ren

Publications and source records attributed to Jingchuang Ren.

3 recordsLinked to original sources

arXiv2024

Fujita phenomena in nonlinear fractional Rayleigh-Stokes equations

This paper concerns the Cauchy problems for the nonlinear Rayleigh-Stokes equation and the corresponding system with time-fractional derivative of order $α\in(0,1)$, which can be used to simulate the anomalous diffusion in viscoelastic fluids. It is shown that there exists the critical Fujita exponent which separates systematic blow-up of the solutions from possible global existence, and the critical exponent is independent of the parameter $α$. Different from the general scaling argument for parabolic problems, the main ingredients of our proof are suitable decay estimates of the solution operator and the construction of the test function.

math.AP↗

arXiv2023

Mean Field Games with infinitely degenerate diffusion and non-coercive Hamiltonian

In this paper, we consider a class of infinitely degenerate partial differential systems to obtain the Nash equilibria in the mean field games. The degeneracy in the diffusion and the Hamiltonian may be different. This feature brings difficulties to the uniform boundness of the solutions, which is central to the existence and regularity results. First, from the perspective of the value function in the stochastic optimal control problems, we prove the Lipschitz continuity and the semiconcavity for the solutions of the Hamilton-Jacobi equations (HJE). Then the existence of the weak solutions for the degenerate systems is obtained via a vanishing viscosity method. Furthermore, by constructing an auxiliary function, we conclude the regularity of the viscosity solution for the HJE in the almost everywhere sense.

math.AP↗

arXiv2023

Degenerate Mean Field Games with Hörmander diffusion

In this paper, we study a class of degenerate mean field game systems arising from the mean field games with Hörmander diffusion, where the generic player may have a ``forbidden'' direction at some point. Here we prove the existence and uniqueness of the classical solutions in weighted Hölder spaces for the PDE systems, which describe the Nash equilibria in the games. The degeneracy causes the lack of commutation of vector fields and the fundamental solution which are the main difficulties in the proof of the global Schauder estimate and the weak maximum principle. Based on the idea of the localizing technique and the local homogeneity of degenerate operators, we extend the maximum regularity result and obtain the global Schauder estimates. For the weak maximum principle, we construct a subsolution instead of the fundamental solution of the degenerate operators.

math.AP↗