Journal of Hyperbolic Differential Equations

We investigate the uniqueness of entropy solution to 2D Riemann problem of compressible isentropic Euler system with maximum density constraint. The constraint is imposed with a singular pressure. Given initial data for which the standard self-similar solution consists of one shock or one shock and one rarefaction wave, it turns out that there exist infinitely many admissible weak solutions. This extends the result of Markfelder and Klingenberg in [S. Markfelder and C. Klingenberg, The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock, Arch. Ration. Mech. Anal. 227(3) (2018) 967–994] for classical Euler system to the case with maximum density constraint. Also some estimates on the density of these solutions are given to describe the behavior of solutions near congestion.

We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus $T=R/2\pi Z:{\partial }_{t}u-i{\partial }_x^{2}u=\alpha {\partial }_x(|u{|}^{2}u)+\beta {\partial }_x[H(|u{|}^2)u]$ for $(t,x)\in [0,T]\times T$, where the constants $\alpha ,\beta$ are such that $\alpha \in R$ and $\beta <0$, and $H$ denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces $H^s$ for $s>3/2$. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when $\beta =0$, cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem.

In this paper, we shall prove local and global well-posedness of the Cauchy problem for small initial data in $H^s(T)$, $s>1/2$. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term $\beta {\partial }_x[H(|u{|}^2)u]$, in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain forward-in-time regularization and backward-in-time ill-posedness results.

In this paper, we study the blowup and non-global existence of smooth solutions to the one-dimensional Euler–Boltzmann equations of radiation hydrodynamics. First, we improve the blowup result in [P. Jiang and Y. G. Wang, Initial-boundary value problems and formation of singularities for one-dimensional non-relativistic radiation hydrodynamic equations, J. Hyperbolic Differential Equations 9 (2012) 711–738] on the half line $[0,+\infty )$ for large initial data by removing a restrict condition. Next, we obtain a new blowup result on the half line $[0,+\infty )$ by introducing a new momentum weight. Finally, we present two non-global existence results for the smooth solutions to the one-dimensional Euler–Boltzmann equations with vacuum on the interval $[0,1]$ by introducing some new average quantities.

We prove via convex integration a result that allows to pass from a so-called subsolution of the isentropic Euler equations (in space dimension at least 2) to exact weak solutions. The method is closely related to the incompressible scheme established by De Lellis–Székelyhidi, in particular, we only perturb momenta and not densities. Surprisingly, though, this turns out not to be a restriction, as can be seen from our simple characterization of the $\mathrm{\Lambda }$-convex hull of the constitutive set. An important application of our scheme has been exhibited in recent work by Gallenmüller–Wiedemann.

Conservation laws are well known to be a crucial part of modeling. Considering such models with the inclusion of non-local flows is becoming increasingly important in many models. On the other hand, kinetic equations provide interesting theoretical results and numerical schemes for the usual conservation laws. Therefore, studying kinetic equations associated to conservation laws for non-local flows naturally arises and is very important. The aim of this paper is to propose kinetic models associated to conservation laws with a non-local flux in dimension $d$ and to prove the existence of solutions for these kinetic equations. This is the very first result of this kind. In order for the paper to be as general as possible, we have highlighted the properties that a kinetic model must verify in order that the present study applies. Thus, the result can be applied to various situations. We present two sets of properties on a kinetic model and two different techniques to obtain an existence result. Finally, we present two examples of kinetic model for which our results apply, one for each set of properties.

We show global existence backward from scattering data at infinity for semilinear wave equations satisfying the null condition or the weak null condition. Semilinear terms satisfying the weak null condition appear in many equations in physics. The scattering data is given in terms of the radiation field, although in the case of the weak null condition there is an additional logarithmic term in the asymptotic behavior that has to be taken into account. Our results are sharp in the sense that the solution has the same spatial decay as the radiation field does along null infinity, which is assumed to decay at a rate that is consistent with the forward problem. The proof uses a higher order asymptotic expansion together with a new fractional Morawetz estimate with strong weights at infinity.

We consider a $2\times 2$ system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different end-states as $x\to \pm \infty$. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as $x\to \pm \infty$, we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays.

We consider the Cauchy problems in $R^n$ for the wave equation with a weighted $L^1$-initial data. We derive sharp infinite time blowup estimates of the $L^2$-norm of solutions in the case of $n=1$ and $n=2$. Then, we apply it to the local energy decay estimates for $n=2$, which is not studied so completely when the $0$th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from a technique used in [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ. 257 (2014) 2159–2177; R. Ikehata and M. Onodera, Remarks on large time behavior of the $L^2$-norm of solutions to strongly damped wave equations, Differ. Integral Equ. 30 (2017) 505–520].