We establish the global well-posedness of weak solutions to the Cauchy problem for a broad class of multi-dimensional $n\times n$ Keyfitz-Kranzer type systems with homogeneous flux. Our proof relies on a novel viscous approximation that yields the necessary compactness estimates. By exploiting the specific structure of this proposed approximation, we derive a priori uniform $L^{\infty}$ bounds and prove the $L^1_{loc}$ precompactness of the sequence of approximate solutions. This framework allows us to rigorously justify the vanishing viscosity limit. Furthermore, under suitable assumptions on the initial data, we employ the relative entropy method to analyze the $L^2$-time decay of large perturbations of the viscous shock, up to a dynamical shift.
In this paper, which originates from Chapter 4 of the author's PhD thesis [1, Chap. 4], we investigate the global well-posedness of a class of partially diffusive hyperbolic systems in critical homogeneous Besov spaces. Our main objective is to extend the analysis developed in our previous work [2] to a functional fram...
In this paper, we consider $2\times2$ hyperbolic systems of conservation laws in one spatial dimension. We use the computational ansatz introduced in the hyperbolic theory by the author and Sz\'ekelyhidi to study the constitutive set corresponding to the PDE. The rank-one convex geometry of this set relates to non-uniq...
We establish local existence and uniqueness of solutions for the ideal incompressible magnetohydrodynamics system posed on $[0,T]\times\mathbb{R}^n$, $n\ge2$, with a nonzero constant initial magnetic field $\mathbf{B}_0$ and arbitrary divergence-free velocity data $v_0\in H^s$, in the range $(n+1)/2<s\le n/2+1$. The pr...
For weak solutions to quasilinear degenerate parabolic equations of $p$-Laplace type, a central obstacle in applying the method of intrinsic scaling to prove their H\"{o}lder regularity is the derivation of forward-in-time propagation estimate for the spatial measure of level sets. In this paper, we revisit and overcom...
It is known that a major difficulty in proving H\"{o}lder regularity for solutions to quasilinear singular parabolic equations of $p$-Laplace type via the method of intrinsic scaling is to establish the decay estimate of the space-time measure of level sets. In this paper, we consider an inhomogeneous singular paraboli...
We study quasilinear symmetrizable partially dissipative hyperbolic systems with non-autonomous relaxation coefficients in $\mathbb{R}^d$ ($d\geq1$). The existence of global strong solutions is established in a critical regularity setting for systems satisfying the so-called Shizuta-Kawashima (SK) and entropy condition...
