Journal of Hyperbolic Differential Equations

We study the Cauchy problem for multi-dimensional compressible relativistic hydrodynamics in the presence of a radiation field. First, based on the theory of quasilinear symmetric hyperbolic, we establish the local existence of smooth solutions for both non-vacuum and vacuum cases. Next, in the spirit of Sideris’ work [T. Sideris, Formation of singularities of solutions to nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 86 (1984) 369–381; T. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985) 475–485], we show that smooth solutions blow-up in finite time if the initial data is compactly supported and large enough. Compared with the previous work, the main difficulties of the first problem lie in two aspects, we must first deal with the source terms relying on radiative quantities, and we also need to solve out the new coefficients matrices under the Lorentz transformation for vacuum case. The second difficulty arises on how to verifying that the smooth solution has finite propagation speed..

Two compressible immiscible fluids in 1D and in the isentropic approximation are considered. The first fluid is surrounded and in contact with the second one. As the Mach number of the first fluid vanishes, we prove the rigorous convergence for the fully nonlinear compressible to incompressible limit of the coupled dynamics of the two fluids. A key role is played by a suitably refined wave front tracking algorithm, which yields precise $BV$, $L^1$ and weak* convergence estimates, either uniform or explicitly dependent on the Mach number.

The compressible Euler–Maxwell two-fluid system arises in the modeling of magnetized plasmas. We first design crucial energy functionals to capture its dissipative structure, which is relatively weaker in comparison with the one-fluid case in the whole space ${ℝ}^3$, due to the nonlinear coupling and cancelation between electrons and ions. Furthermore, with the aid of $L^p({ℝ}^n)$-$L^q({ℝ}^n)$-$L^r({ℝ}^n)$ time-decay estimates, we obtain the $L^1({ℝ}^3)$-$L^2({ℝ}^3)$ decay rate with the critical regularity ($s_c=3$) for the global-in-time existence of smooth solutions, which solves the decay problem left open in [Y. J. Peng, Global existence and long-time behavior of smooth solutions of two-fluid Euler–Maxwell equations, Ann. IHP Anal. Non Linéaire 29 (2012) 737–759].

We study the global decay properties of solutions to the linear wave equation in 1+3 dimensions on time-dependent, weakly asymptotically flat spacetimes. Assuming non-trapping of null geodesics and a local energy decay estimate, we prove that sufficiently regular solutions to this equation have bounded conformal energy. As an application we also show a conformal energy estimate with vector fields applied to the solution as well as a global $L^{\infty }$ decay bound in terms of a weighted norm on initial data. For solutions to the wave equation in these dynamical backgrounds, our results reduce the problem of establishing the classical pointwise decay rate $t^{-3/2}$ in the interior and $t^{-1}$ along outgoing null cones to simply proving that local energy decay holds.

We consider the system of (pure) gravity water waves in any dimension and in a fluid domain with a general bottom geometry. The unique solvability of this problem was established by Alazard–Burq–Zuily [Invent. Math. 198(1) (2014) 71–163] at a low regularity level where the initial surface is $C^{\frac{3}{2}+}$ in terms of Sobolev embeddings; this result allows the existence of free surfaces with unbounded curvature. Our result states that the solutions obtained in the above work depend continuously on initial data in the strong topology in which the solutions are constructed. This establishes a well-posedness result in the sense of Hadamard.

We consider the nonlinear Dirac equation with a power nonlinearity

We consider the strategy of realizing the solution of a Cauchy problem (CP) with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the three-dimensional (3d), linear wave equation ${\square }_{1+3}U=0$ with radial Cauchy data $U(0,x)=\mathrm{\Phi }(x)=\phi (|x|)$, $U_t(0,x)=\mathrm{\Psi }(x)=\psi (|x|)$. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available for the radial symmetric CP. In treating the 3d wave equation, we therefore insist on robust arguments based on energy methods and strong convergence. (In particular, this work does not address what can be established via solution formulae.) Our findings for the 3d wave equation show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. The standard existence result for the linear wave equation guarantees a unique solution in $C([0,T);H^s({ℝ}^3))$ whenever $(\mathrm{\Phi },\mathrm{\Psi })\in H^s\times H^{s-1}({ℝ}^3)$. However, within the constrained framework outlined above, we obtain strictly lower regularity for solutions obtained as limits of exterior solutions. We also show that external Neumann solutions yield better regularity than external Dirichlet solutions. Specifically, for Cauchy data in $H^2\times H^1({ℝ}^3)$, we obtain $H^1$-solutions via exterior Neumann solutions, and only $L^2$-solutions via exterior Dirichlet solutions.