Unique Classical Solution Research Articles

The paradifferential approach to the local well-posedness of some problems in mixture theory in two space dimensions

In this paper, we consider a class of models describing multiphase fluids in the framework of mixture theory. The considered systems, in their more general form, contain both the gradient of a hydrostatic pressure, generated by an incompressibility constraint, and a compressible pressure depending on the volume fractions of some of the different phases. To approach these systems, we propose an approximation based on the Leray projection, which involves the use of a symbolic symmetrizer for quasi-linear hyperbolic systems and related paradifferential techniques. In two space dimensions, we prove the well-posedness of this approximation and its convergence to the unique classical solution to the original system. In the last part, we shortly discuss the three dimensional case.

The Muskat problem and related topics

In the paper, we consider the evolution of the free boundary separating two immiscible viscous fluids with different constant densities in an absolutely rigid solid body and in an elastic skeleton. The motion of the liquids is described by the Stokes equations driven by the input pressure and the force of gravity. For flows in a bounded domain, we prove the existence and uniqueness of classical solutions and emphasize the study of the properties of the moving boundary separating the two fluids.

Unique global solution to the thermally radiative magnetohydrodynamics equations

In this paper, we consider a non-equilibrium diffusion approximation model in thermally radiative magnetohydrodynamics (MHD) which describe the compressible viscous, heat-conducting fluid coupled with the Maxwell equations governing the behavior of the magnetic field with taking into account the radiation effect under the non-local thermal equilibrium case. The system consists of the radiative transfer equation coupled with the compressible MHD equations, and the magnetic diffusion coefficient, viscosity coefficient, heat-conducting coefficient are all allowed to be positive constants. We prove existence, uniqueness and regularity of global-in-time classical solutions to one-dimensional case of this model. Based on the fundamental local existence results and global-in-time a priori estimates, we can establish the global existence of a unique classical solutions with large initial data to the initial-boundary value problems.

Global and almost global existence for general quasilinear wave equations in two space dimensions

For 2-D quasilinear wave equations of the form □u=F(u,Du,D2u), we show that the corresponding Cauchy problem admits a unique global classical solution with small initial data, if both the quadratic nonlinearity and the cubic nonlinearity satisfy the corresponding null conditions. We also prove that the classical solution is almost global, i.e., the lifespan of solution Tε≥exp⁡(cε−2), where ε>0 denotes the amplitude of the initial data and c is a positive constant, provided that the quadratic nonlinearity satisfies the null condition and ∂uβF(0,0,0)=0, β=3,4. These results extend Alinhac's seminal work Alinhac (2001) [9] to the general case in a unified way.

Synchronization control for reaction–diffusion FitzHugh–Nagumo systems with spatial sampled-data

To gain a better understanding the synchronization mechanism of networked neurons, this paper studies the synchronization control for a class of reaction–diffusion FitzHugh–Nagumo systems, associated with a digraph containing at least one directed spanning tree. A novel control method adopting spatial sampling strategies is proposed, in which the control inputs are constructed directly on the spatial means of system state variables on sampling subsets. After discussing the existence and uniqueness of classical solutions, we show analytically that the synchronization of the controlled systems is equivalent to that of the corresponding FitzHugh–Nagumo systems under the corresponding control inputs. Based on the study on general algebraic connectivity, a sufficient condition for the system synchronization is given, together with case simulations to illustrate the effectiveness and potential of the new control method.

Boundedness and global stability of the two-predator and one-prey models with nonlinear prey-taxis

This paper concerns the reaction–diffusion systems modeling the population dynamics of two predators and one prey with nonlinear prey-taxis. We first investigate the global existence and boundedness of the unique classical solution for the general model. Then, we study the global stabilities of nonnegative spatially homogeneous equilibria for an explicit system with type I functional responses and density-dependent death rates for the predators and logistic growth for the prey. Moreover, the convergence rates are also established.

Existence and nonexistence theorems of global weak solutions to degenerate quasilinear wave equations for the elasticity

We prove the existence and nonexistence of global generalized (nonnegative) solutions of the nonlinearly degenerate wave equations vtt=c(|v|s−1v)xx with the nonnegative initial data v0(x) and s>1. These results are extensions of the results in the second author’s paper (Sugiyama, 2016), where the existence and the nonexistence of the unique global classical solution were studied with a threshold on ∫−∞∞v1(x)dx and the non-degeneracy condition v0(x)≥c0>0 on the initial data.

Global wellposedness of magnetohydrodynamics system with temperature-dependent viscosity

The initial boundary value problem of the one-dimensional magneto-hydrodynamics system, when the viscosity, thermal conductivity, and magnetic diffusion coefficients are general smooth functions of temperature, is considered in this article. A unique global classical solution is shown to exist uniquely and converge to the constant state as the time tends to infinity under certain assumptions on the initial data and the adiabatic exponent γ. The initial data can be large if γ is sufficiently close to 1.

Global attractors of some predator–prey reaction–diffusion systems with density-dependent diffusion and time-delays

This paper deals with a two-species and a three-species predator–prey reaction diffusion systems where the diffusion coefficients are density dependent and time-delays are involved in the reaction functions. The diffusion terms are of porous medium type which are degenerate and the boundary conditions are of Neumann type. The aim of the paper is to investigate the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions of the system. This includes the existence of a unique positive classical solution, global attraction of a steady-state solution, and stability or instability of various semitrivial solutions and the positive steady-state solution.

Existence of global solution to a two-species Keller–Segel chemotaxis model

In this paper, we investigate the global existence of nonnegative solutions of a two-species Keller–Segel model with Lotka–Volterra competitive source terms. By raising the regularity of a solution from [Formula: see text] to [Formula: see text], the existence and uniqueness of the classical global in time solution to this chemotaxis model is proved for any chemotactic coefficients [Formula: see text] when the space dimension is one. Furthermore, it is shown that the model has a unique classical global solution in two and three space dimensions if the chemotactic coefficients [Formula: see text] and [Formula: see text] are small as compared to the diffusion coefficient [Formula: see text] of the chemoattractant.

Global well-posedness of the 3D non-isothermal compressible fluid model of Korteweg type

The aim of this work is to study the global existence and large-time behavior of solutions to the non-isothermal model of capillary compressible fluids derived by Dunn and Serrin (1985). It is proved that the three-dimensional non-isothermal compressible Navier–Stokes–Korteweg system admits a unique global classical solution, provided that the initial energy is suitably small. This result improves previous results obtained by Hattori and Li (1996), where the existence of global classical solutions is established for the initial data close to an equilibrium in some Sobolev space Hs(s≥3). Furthermore, the large time behavior of the solution is also investigated.

Initial boundary value problem for the 3D quasilinear hyperbolic equations with nonlinear damping

ABSTRACTWe investigate global existence and asymptotic behavior of the 3D quasilinear hyperbolic equations with nonlinear damping on a bounded domain with slip boundary condition, which describes the propagation of heat waves for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of classical solutions are obtained when the initial data are near its equilibrium. Time asymptotically, the internal energy is conjectured to satisfy the porous medium equation and the heat flux obeys the classical Darcy’s-type law. Based on energy estimates, we show that the classical solution converges to steady state exponentially fast in time. Moreover, we also verify that the same is true for the corresponding initial boundary value problem of porous medium equation and thus justifies the validity of Darcy’s-type law in large time.

Rumor diffusion model with spatio-temporal diffusion and uncertainty of behavior decision in complex social networks

In this paper, a rumor diffusion model with uncertainty of human behavior under spatio-temporal diffusion framework is established. Take physical significance of spatial diffusion into account, a diffusion threshold is set under which the rumor is not a trend topic and only spreads along determined physical connections. Heterogeneity of degree distribution and distance distribution has also been considered in theoretical model at the same time. The global existence and uniqueness of classical solution are proved with a Lyapunov function and an approximate classical solution in form of infinite series is constructed with a system of eigenfunction. Simulations and numerical solutions both on Watts–Strogatz (WS) network and Barabási–Albert (BA) network display the variation of density of infected connections from spatial and temporal dimensions. Relevant results show that the density of infected connections is dominated by network topology and uncertainty of human behavior at threshold time. With increase of social capability, rumor diffuses to the steady state in a higher speed. And the variation trends of diffusion size with uncertainty are diverse on different artificial networks.

Global classical solutions of compressible isentropic Navier–Stokes equations with small density

This paper concerns the Cauchy problem of compressible isentropic Navier–Stokes equations in the whole space R3. First, we show that if ρ0∈Lγ∩H3, then the problem has a unique global classical solution on R3×[0,T] with any T∈(0,∞), provided the upper bound of the initial density is suitably small and the adiabatic exponent γ∈(1,6). If, in addition, the conservation law of the total mass is satisfied (i.e., ρ0∈L1), then the global existence theorem with small density holds for any γ>1. It is worth mentioning that the initial total energy can be arbitrarily large and the initial vacuum is allowed. Thus, the results obtained particularly extend the one due to Huang–Li–Xin (Huang et al., 2012), where the global well-posedness of classical solutions with small energy was proved.

The modified Camassa-Holm equation in Lagrangian coordinates

In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data $m_0$, we show that classical solutions to this equation blow up in finite time $T_{max}$. Before $T_{max}$, existence and uniqueness of classical solutions are established. Lifespan for classical solutions is obtained: $T_{max}\geq \frac{1}{||m_0||_{L^\infty}||m_0||_{L^1}}.$ And there is a unique solution $X(\xi,t)$ to the Lagrange dynamics which is a strictly monotonic function of $\xi$ for any $t\in[0,T_{max})$: $X_\xi(\cdot,t)>0$. As $t$ approaching $T_{max}$, we prove that classical solution $m(\cdot ,t)$ in Eulerian coordinate has a unique limit $m(\cdot,T_{max})$ in Radon measure space and there is a point $\xi_0$ such that $X_\xi(\xi_0,T_{max})=0$ which means $T_{max}$ is an onset time of collision of characteristics. We also show that in some cases peakons are formed at $T_{max}$. After $T_{max}$, we regularize the Lagrange dynamics to prove global existence of weak solutions $m$ in Radon measure space.

On the dynamics of finite temperature trapped Bose gases

The system that describes the dynamics of a Bose–Einstein Condensate (BEC) and the thermal cloud at finite temperature consists of a nonlinear Schrodinger (NLS) and a quantum Boltzmann (QB) equations. In such a system of trapped Bose gases at finite temperature, the QB equation corresponds to the evolution of the density distribution function of the thermal cloud and the NLS is the equation of the condensate. The quantum Boltzmann collision operator in this temperature regime is the sum of two operators C12 and C22, which describe collisions of the condensate and the non-condensate atoms and collisions between non-condensate atoms. Above the BEC critical temperature, the system is reduced to an equation containing only a collision operator similar to C22, which possesses a blow-up positive radial solution with respect to the L∞ norm (cf. [29]). On the other hand, at the very low temperature regime (only a portion of the transition temperature TBEC), the system can be simplified into an equation of C12, with a different (much higher order) transition probability, which has a unique global classical positive radial solution with weighted L1 norm (cf. [3]). In our model, we first decouple the QB, which contains C12+C22, and the NLS equations, then show a global existence and uniqueness result for classical positive radial solutions to the spatially homogeneous kinetic system. Different from the case considered in [29], due to the presence of the BEC, the collision integrals are associated to sophisticated energy manifolds rather than spheres, since the particle energy is approximated by the Bogoliubov dispersion law. Moreover, the mass of the full system is not conserved while it is conserved for the case considered in [29]. A new theory is then supplied.

The dividend problem with a finite horizon

We characterise the value function of the optimal dividend problem with a finite time horizon as the unique classical solution of a suitable Hamilton–Jacobi–Bellman equation. The optimal dividend strategy is realised by a Skorokhod reflection of the fund’s value at a time-dependent optimal boundary. Our results are obtained by establishing for the first time a new connection between singular control problems with an absorbing boundary and optimal stopping problems on a diffusion reflected at $0$ and created at a rate proportional to its local time.

Existence and Uniqueness Theorem for a Model of Bimolecular Surface Reactions

We prove the existence and uniqueness of classical solutions to a coupled system of parabolic and ordinary differential equations in which the latter are determined on the boundary. This system describes a model of bimolecular surface reaction between carbon monoxide and nitrous oxide running on supported rhodium in the case of slow desorption of the products.

Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production

We study the Neumann initial-boundary problem for the chemotaxis system \left\{ \begin{array}{ll} u_t= \Delta u - \nabla \cdot (u\nabla v), & x\in \Omega, \, t>0, \\ 0=\Delta v - \mu(t)+w, & x\in \Omega, \, t>0, \\ \tau w_t + \delta w = u, & x\in \Omega, \, t>0, \end{array} \right. \qquad \qquad (\star) in the unit disk \Omega:=B_1(0)\subset R^2 , where \delta\ge 0 and \tau>0 are given parameters and \mu(t):=\frac{1}{|\Omega|} \int_\Omega w(x,t)dx , t>0 . It is shown that this problem exhibits a novel type of critical mass phenomenon with regard to the formation of singularities, which drastically differs from the well-known threshold property of the classical Keller-Segel system, as obtained upon formally taking \tau\to 0 , in that it refers to blow-up in infinite time rather than in finite time. Specifically, it is first proved that for any sufficiently regular nonnegative initial data u_0 and w_0 , ( \star ) possesses a unique global classical solution. In particular, this shows that in sharp contrast to classical Keller-Segel-type systems reflecting immediate signal secretion by the cells themselves, the indirect mechanism of signal production in ( \star ) entirely rules out any occurrence of blow-up in finite time. However, within the framework of radially symmetric solutions it is next proved that \|u(\cdot,t)\|_{L^\infty(\Omega)} \to \infty \qquad \text{as } t\to\infty.

Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs

In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn et al. (2014) to BSDEs, the existence and the uniqueness of the solution (Yt,ξ,Zt,ξ,Ht,ξ), (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) of the split equations are proved. The first and the second order derivatives of the process (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) with respect to x, the derivative of the process (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) with respect to the measure Pξ, and the derivative of the process (∂μYt,x,Pξ(y),∂μZt,x,Pξ(y),∂μHt,x,Pξ(y)) with respect to y are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in L2. The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function V(t,x,Pξ)≔Ytt,x,Pξ is regular and allows to show with the help of a new Itô formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type.