Related papers: On hyperbolic systems with time dependent H\"older…
We study a class of parabolic equations having first order terms with superlinear (and subquadratic) growth. The model problem is the so-called viscous Hamilton-Jacobi equation with superlinear Hamiltonian. We address the problem of having…
This paper investigates the inverse problems of determining a space-dependent source for thermoelastic systems of type III under adequate time-averaged or final-in-time measurements and conditions on the time-dependent part of the sought…
Statistical mechanics for states with complex eigenvalues, which are described by Gel'fand triplet and represent unstable states like resonances, are discussed on the basis of principle of equal ${\it a priori}$ probability. A new entropy…
Motivated by recent results on the (possibly conditional) regularity for time-dependent hypoelliptic equations, we prove a parabolic version of the Poincar\'e inequality, and as a consequence, we deduce a version of the classical Moser…
We study well posedness of time--dependent Hamilton--Jacobi equations on a network, coupled with a continuous initial datum and a flux limiter. We show existence and uniqueness of solutions as well as stability properties. The novelty of…
This paper concerns with the large time behavior of solutions to a diffusion approximation radiation hydrodynamics model when the initial data is a small perturbation around an equilibrium state. The global-in-time well-posedness of…
In this paper, a class of systems of pseudo-parabolic PDEs is considered. These systems (S)$_\varepsilon$ are derived as a pseudo-parabolic dissipation system of Kobayashi--Warren--Carter energy, proposed by [Kobayashi et al., Physica D,…
Using Zvonkin's transform and the Poisson equation in $R^d$ with a parameter, we prove the averaging principle for stochastic differential equations with time-dependent H\"older continuous coefficients. Sharp convergence rates with order…
This paper is devoted to studying the Hamilton-Jacobi-Bellman equations with distribution-valued coefficients, which is not well-defined in the classical sense and shall be understood by using paracontrolled distribution method introduced…
We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Friedlin--Wentzell type random system for times that are rather long, but…
The paper is concerned with the effect of the spatio-temporal heterogeneity on the principal eigenvalue of some linear time-periodic parabolic system. Various asymptotic behaviors of the principal eigenvalue and its monotonicity, as a…
These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of…
Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We,…
We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on…
We consider the temporal periodic solutions to general nonhomogeneous quasilinear hyperbolic equations with a kind of weak diagonal dominant structure. Under the temporal periodic boundary conditions, the existence, stability and uniqueness…
This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s(\mathbb{R}^2)$ for $s…
Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H \subseteq V^*$: \begin{align*} \left\{ \begin{aligned} dX(t) & = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in…
We study the behavior of shallow water waves over periodically-varying bathymetry, based on the first-order hyperbolic Saint-Venant equations. Although solutions of this system are known to generally exhibit wave breaking, numerical…
We study the initial value problem for the wave equation and the ultrahyperbolic equation for data posed on initial surface of mixed signature (both spacelike and timelike). Under a nonlocal constraint, we show that the Cauchy problem on…
The Cauchy problem for the Chern-Simons-Higgs system in the (2+1)-dimensional Minkowski space in temporal gauge is locally well-posed for low regularity initial data improving a result of Huh. The proof uses the bilinear space-time…