This is a survey of recent results concerning the mathematical theory of compressible, viscous, and heat conducting fluids. Starting from the basic physical principles, notably the First and Second laws of thermodynamics, we introduce a concept of weak solutions to complete fluid systems and analyze their asymptotic behavior. In particular, the long time behavior and scale analysis will be performed. We also introduce a new concept of relative entropy for the system and show how it can be used in the problem of weak-strong uniqueness and the inviscid limits.

For three interesting kinetic models of clustering, we review results on dynamical phenomena related to the approach to self-similar form and their close connections to probability theory. For Smoluchowski's coagulation equation with additive rate kernel, we describe the scaling attractor and show how dynamics on it is trivialized in terms of Bertoin's Lévy–Khintchine-like representation. For a model motivated by domain coarsening dynamics in the Allen–Cahn equation, we describe the remarkable solution procedure found by Gallay and Mielke, and the ensuing classification of domains of attraction for self-similar solutions. And we describe the rigorous connection between Smoluchowski's equation and random shock coalescence in the inviscid Burgers equation. Recent work of Menon indicates that the latter problem is completely integrable for initial data comprising a spatial Markov process.

The FitzHugh–Nagumo system is a well-known reaction-diffusion model for exhibiting self-organized patterns. Besides regular patterns found in a neighborhood of Turing's instability, localized structures are also observed in experiment and numerical simulation. In particular, fronts and pulses are the most well-known localized structures in reaction-diffusion systems. This article is aimed at some recent results on the variational approach for studying standing waves of FitzHugh–Nagumo system.

It is known that the Green–Naghdi equations are higher order approximations for the water wave problem in a shallow water regime. We derive corresponding equations in the case of moving bottom as a mode of tsunamis caused by the slow deformation of the bottom, and give a mathematically rigorous justification of the model.

We consider the Cauchy problem for the two dimensional nonlinear Klein–Gordon equation with a quadratic nonlinearity. In the present paper we find more natural conditions for the initial data than those of previous works to ensure the existence of scattering states.

In this paper we present the recent existence results from [14], [15] on weak solutions to the the steady Navier–Stokes equations for three-dimensional compressible isentropic flows with large data for any specific heat ratio $\gamma \gt 1$. The existence is proved in the framework of the weak convergence method due to Lions [16] by establishing a new a priori potential estimate of both pressure and kinetic energy (in a Morrey space) and using a bootstrap argument. The results presented in the current paper extend the existence of weak solutions in [9] from $\gamma \gt 4/3$ to $\gamma \gt 1$.

Fully nonlinear second-order uniformly elliptic partial differential equations (PDE for short) with unbounded ingredietns are considered. The Aleksandrov–Bakelman–Pucci (ABP for short) maximum principle for $L^p$-viscosity solutions of fully nonlinear, second-order uniformly elliptic PDE are shown.

The results here are joint works with A. Świȩch in [12], [13], [14], [15].

We survey the recent results for the damped wave equations and obtain the critical exponent for the semilinear problem. Those results are based on the diffusion phenomenon of the damped wave equation.

The time-evolution equations for an isolated domain in an excitable reaction-diffusion system are derived both in two and three dimensions by an interfacial approach near the drift bifurcation where a motionless state becomes unstable and a domain starts propagation at a certain velocity. The coupling between shape deformation of domain and the migration velocity is taken into consideration. When the relaxation of shape deformation is slow enough, a straight motion becomes unstable and several kinds of motion of domain appear depending on the parameters. The self-propelled domain dynamics under the external fields is also studied.

We give an asymptotic formulas in time of solutions to the linear Schrödinger equation with third and fourth order dispersions in one space dimension. Then we apply those formulas to drive the long time behavior of solution to the third and the fourth order Schrödinger type equations with the cubic nonlinearity.

We survey results on shapes of the stable steady states of two nonlinear problems: a variational problem with a mass constraint and the shadow system of activator-inhibitor type. We see that the stable steady states of the two problems are the Morse index one solutions of a scalar reaction-diffusion equation. We study shapes of the Morse index one solutions and see that the shapes of the Morse index one solutions are deeply related to the "hot spots" conjecture of J. Rauch. We also survey results on the "hot spots" conjecture and related problems.

Let us consider the Liouville equation $$ -\Delta u = \lambda V(x) e^u \quad\text{in}\ \Omega,\quad u = 0\quad \text{on}\ \partial\Omega, $$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$, $V(x) \gt 0$ is a given function in $C^1(\overline{\Omega})$, and $\lambda \gt 0$ is a constant. Let $\{u_n\}$ be an $m$-point blowing up solution sequence for $\lambda = \lambda_n \downarrow 0$, in the sense that $$ \lambda_n\int_\Omega V(x)e^{u_n} dx \to 8\pi m\quad \text{as}\ n\to\infty $$ for $m\in\mathbb{N}$. We prove that the number of blow up points $m$ is less than or equal to the Morse index of $u_n$ for $n$ sufficiently large. This extends the main result of the recent paper [13] to an inhomogeneous ($V \not\equiv 1$) case.

This note is devoted to reviewing the authors' recent work [1] on the stability analysis of asymptotic profiles of solutions to the Cauchy–Dirichlet problem for the fast diffusion equation.

In this paper, we are interested in a motion of non-closed planar curves with infinite length. The motion is governed by a gradient flow for the modified one-dimensional Willmore functional. We shall prove a long time existence of solution of an initial boundary value problem for the flow.

Systems of linear diffusion equations are considered in smooth bounded domains under nonlinear boundary conditions of Neumann type. We show that the interaction between the difference in diffusion rates and the nonlinear boundary conditions destabilizes uniform steady states, resulting in time periodic spatial patterns.

In this article, we consider the Eldar model [3] from embryology in which a bone morphogenic protein, a short gastrulation protein, and their compound react and diffuse. We carry out a perturbation analysis in the limit of small diffusivity of the bone morphogenic protein. This analysis establishes conditions under which some elementary results of [3] are valid.

Using the variational method Chenciner and Montgomery proved the existence of a new periodic solution of figure-eight shape to the planar three-body problem. Since then, a number of periodic solutions have been discovered as minimizers. We present a minimax approach to the $n$-body problem and prove the existence of some periodic solutions as minimax points of the action functional.

We study bifurcations of homoclinic orbits to hyperbolic saddles and saddle-centers in reversible systems analytically by Melnikov-type methods and algebraically by differential Galois theory.

We investigate the large-time behavior of viscosity solutions of Hamilton–Jacobi equations with noncoercive Hamiltonian in a multi-dimensional Euclidean space. Our motivation comes from a model describing growing faceted crystals recently discussed by E. Yokoyama, Y. Giga and P. Rybka (Phys. D, 237 (2008), no. 22, 2845–2855). We prove that the average growth rate of a solution is constant only in a subset, which will be called effective domain, of the whole domain and give the asymptotic profile in the subset. This means that the large-time behavior for noncoercive problems may depend on the space variable in general, which is different from the usual results under the coercivity condition. Moreover, on the boundary of the effective domain, the gradient with respect to the $x$-variable of solutions blows up as time goes to infinity. Therefore, we are naturally led to study singular Neumann problems for stationary Hamilton–Jacobi equations. We establish the existence and comparison results for singular Neumann problems and apply the results for a large-time asymptotic profile on the effective domain.

This paper introduces a linear scheme to approximate the solutions of the general nonlinear cross-diffusion system. After discretizing the scheme in space, we obtain a versatile, easy to implement and stable numerical scheme for the cross-diffusion system. Numerical experiments are carried out to examine rates of convergence with respect to the time step and the spatial mesh sizes.

We will sketch a curvature adjusted method for a moving plane curves and apply it to the Hele-Shaw flow in a time-dependent gap together with the boundary element method.

The note discusses the motion of solitary waves on the free surface of a layer of water. The rigorous results for the existence of solitary-wave solutions of exact governing equations are given. To generate such surface waves, a moving bump placed at the bottom or a pressure source on the free surface is used. A model equation, called forced Korteweg–de Vries (FKdV) equation, is numerically studied and multi- solitary- wave solutions are obtained. Then, the numerical solutions are compared with experimental results using a water tank with a moving bump at the bottom.

We consider nonlinear Schrödinger equation with nonlocal nonlinearity which is described by a growing interaction potential. This model contains low-dimensional Schrödinger–Poisson system. We briefly survey recent progress on this subject and then show existence of ground state in a specific model.

The aim of this note is to give an overview concerning the mixed problem for a system of nonlinear wave equations with small and smooth initial data. In particular, we are interested in the three and two space dimensional case.

The Cauchy problem for nonlinear wave equations with localized dissipation is considered in exterior domains outside of compact obstacles in three spatial dimensions. Under the null conditions for the quadratic nonlinear terms, the global solutions are shown for sufficiently small data. The solutions which have different propagation speeds are considered.

This article is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima–Shizuta stability condition formulated in [8], [4]. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (cf. the Timoshenko system and the Euler–Maxwell system). Therefore our purpose of this paper is to formulate a new structural condition and to analyze the weak dissipative structure for general systems with non-symmetric relaxation. This article is a survey of the paper [5].

In this note we provide some precise estimates explaining the diffusive structure of partially dissipative systems with time-dependent coefficients satisfying a uniform Kalman rank condition. Precisely, we show that under certain (natural) conditions solutions to a partially dissipative hyperbolic system are asymptotically equivalent to solutions of a corresponding parabolic equation.

The approach is based on an elliptic WKB analysis for small frequencies in combination with exponential stability for large frequencies due to results of Beauchard and Zuazua and arguments of perturbation theory.

In this article, we will give a reciprocity formula for generalized Gauss sums and its application to Schrödinger equations on certain compact symmetric spaces.

It is well known that the vector fields $$ \Omega = x\wedge D = (\Omega_{ij})_{i<j},\qquad \Omega_{ij} = x_i D_j - x_j D_i $$ commute with the Laplacian $-\Delta$. Hence we have $$ Pu = f \quad\Rightarrow\quad P(\Omega u) = \Omega f, $$ where $P$ is a function of $-\Delta$, and in this way we can control the growth/decaying order of solution $u$ to the equation $Pu = f$. This fact was actually used to induce some decaying estimates for the wave equation ([3]) in a context of nonlinear analysis, and smoothing estimates for the Scrödinger equation ([6]) in a critical case. In this article, we will discuss how to trace this idea for equations with the Laplacian $-\Delta$ replaced by general elliptic (pseudo-)differential operators.

We present a study of dissipative structures for a class of the coupled kinetic-fluid models with partial relaxations at the linearized level. It is a generalization of several known results in the decoupled case that is either for the kinetic model or for the symmetric hyperbolic system. Precisely, a partially dissipative linearized system is of the type $(p, q)$ if the real parts of all eigenvalues in terms of the frequency variable $k$ admit an upper bound $-|k|^{2p}/(1 + |k|^2)^q$ up to a common positive constant. It is well known that a symmetric hyperbolic system with partial relaxation is of the type $(1, 1)$ if the so-called Shizuta–Kawashima conditions are satisfied. In the current study of the coupled kinetic-fluid models, we postulate more general conditions together with some concrete examples to include the case $(1, 2)$ investigated also in [14] and the new case $(2, 3)$. Thus, the coupled kinetic-fluid models may exhibit more complex dissipative structures.

We study $C_0$-semigroups acting on a Banach space which possess an invariant property with respect to an action of the multiplicative group of positive real numbers (scaling). Some properties on the domains or the spectrum of the associated generators are presented.

This work is concerned with smooth periodic solutions for the compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force. The existence and uniqueness of the global smooth solution is established by using energy method.

We examine a two-component reaction-diffusion system on the real axis with quadratic nonlinearity. Using semigroup estimates, we obtain a solution to our nonlinear system for long-time. For appropriate initial data, we show that a slowly-varying, scaled solution of the Allen–Cahn equation will estimate the solution of our nonlinear system for long-time. We additionally extend this work to $\mathbb{R}^d$.

We investigate decay properties of solutions to the linearized compressible Navier–Stokes equation around time-periodic parallel flow. We show that if the Reynolds and Mach numbers are sufficiently small, solutions of the linearized problem decay in $L^2$ norm as an $n-1$ dimensional heat kernel. Furthermore, we prove that the asymptotic leading part of solutions is given by solutions of an $n-1$ dimensional linear heat equation with a convective term multiplied by time-periodic function.

In this paper, we consider a class of second order hyperbolic systems of viscoelasticity in the whole space and show the global existence and sharp decay estimates of solutions under the smallness condition on the initial data. The tools that we use here are the time-weighted energy method and the semigroup argument.

We outline some results on the existence of standing-wave solutions to a coupled non-linear Klein–Gordon equation. Standing-waves are obtained as minimizers of the energy under a two-charges constraint. The ground state is stable. The standing-waves are stable provided a non-degeneracy condition is satisfied.

We summarize our recent results on the low Mach number limit for the compressible non-isentropic magnetohydrodynamic equations. We consider two cases: (i) small variations on density and temperature with well-prepared initial data; (ii) large variations on density and temperature with ill-prepared initial data. In both cases, we establish the limits rigorously.

In this paper, we consider radially symmetric positive stationary solutions for the competition-diffusion system which describes the dynamics of population for a two-competing-species community, and discuss the bifurcation structure of solution by employing the comparison principle and the bifurcation theory.

We introduce the wave front set $WF_s^{p,q}$ by using the wave packet transform. This is another characterization of the Fourier–Lebesgue type wave front set $WF_{\mathcal{F}L_s^q}$. We apply this to the propagation of singularities for the wave equation.

We show $L^p - L^q$ estimates with the $\langle x\rangle^s$ type weight of Stokes semigroup in exterior domains and perturbed half-spaces. Moreover as application of these estimates, we obtain weighted estimates for global solution to the Navier–Stokes equations with small data in these domains.

The initial value problem of a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions is studied. We show there exists a unique global solution for this initial value problem which decays like $t^{-1}$ as $t\to +\infty$ in $\mathbf{L}^\infty (\mathbb{R}^2)$ for small initial data in lower order Sobolev spaces.

In this paper, we investigate the existence, uniqueness and stability of the ground states of nonlinear Schrödinger type equations with a small fourth order dispersion. Such equations appear in the higher order approximation of the propagation of laser beam in Kerr medium. We show that for the critical case, the ground state, which is unstable in the absence of the fourth order dispersion, becomes stable with small fourth order term.

This note is concerned with Strichartz estimates for solutions to Schrödinger equations with asymptotically flat metrics and subquadratically growing potentials. We prove the estimates outside a large compact set centered at origin. Under the non-trapping condition, we also prove global-in-space estimates.

The Cauchy problem for a semilinear heat equation $$ w_t = \Delta w + w^p\quad \text{in}\ \mathbf{R}^N \times (0,\infty) $$ with singular initial data $w(x,0) =\lambda a (x/|x|) |x|^{-2/(p-1)}$ for $x\in\mathbf{R}^N\setminus \{0\}$ is studied, where $N \gt 2$, $p=(N+2)/(N-2)$, $\lambda \gt 0$ is a parameter, and $a\ge0$, $a\not\equiv0$. We investigate the asymptotic properties of the profile of positive self-similar solutions to the problem as $\lambda\to0$ when $N = 3,4,5$.

Spherically symmetric gravitational collapse of a scalar field in the Einstein–Gauss–Bonnet theory will be considered. We show some global results for this system, which supports the validity of the cosmic censorship conjecture.

We consider the dynamics of an oscillatory pulse (standing breather, SB) of front-back type, in which the motion of two interfaces that interact through a continuous field is described by a mixed ODE-PDE system. We carry out a center manifold reduction around the Hopf singularity of a stationary pulse solution, which provides us insight into the underlying mechanism for a sliding motion of SB in a jump-type spatial heterogeneous medium.

In plasma physics, the Bohm criterion is known as a necessary condition for the formation of the boundary layer called sheath. Our aim is to validate the Bohm criterion from the mathematical point of view. First, we define the sheath as a stationary solution to the Euler–Poisson equations over a half space since the Bohm criterion is proved to be sufficient for the existence of the stationary solution. Then we study the asymptotic stability of the stationary solution under the degenerate or nondegenerate Bohm criterion.

We determine the range of Sobolev regularity for the Maxwell–Dirac system in $1+1$ space time dimensions to be well-posed locally. The well-posedness follows from the null form estimates. Outside the range for the well-posedness, we show either the flow map is not continuous or not twice differentiable at zero.

This paper is devoted to the study of the initial value problem for some semilinear damped beam equation which has a non-negative definite energy functional. We prove a uniform bound and decay estimates of a solution for the equation with large initial data using the method introduced by Kawashima–Nakao–Ono [3].

In this paper we consider a free-boundary problem for the spherically symmetric flow of a self-gravitating viscous and heat-conductive gas over a rigid sphere. Considering the problem in the Lagrangian-mass coordinate system, we investigate the temporally global existence and behaviour of the solution. Especially, we focus our discussion on the corresponding stationary problem, whose solution gives an asymptotic profile of the solution of the original (time-dependent) problem.

In this work, we show the existence of traveling wave solution of a diffusive predator-prey model with Holling type III functional response. The analysis is based on Wazewski's principle in the four-dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator-prey system under the moving coordinates.