Global weak solutions and breaking waves to the Degasperis–Procesi equation with linear dispersion

We study here an initial-value problem for the Degasperis–Procesi equation with a strong dispersive term, which is an approximation to the incompressible Euler equations for shallow water waves. We first determine the blow-up set of breaking waves to the equation. We then prove the existence and uniqueness of global weak solutions to the equation with certain initial profiles.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

Structure of the set of stationary solutions of phase-lock equations in superconductivity

In previous article [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075], we introduced a system of equations (phase-lock equations) to model the superconductivity phenomena. We investigated its connection to Ginzburg–Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the steady-state problem associated with the phase-lock equations. We prove that the steady-state problem has multiple solutions and show that the solution set enjoys some structural properties as proved by Foias and Teman for the Navier–Stokes equations in [C. Foias, R. Teman, Structure of the set of stationary solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. XXX (1977) 149–164].     

On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis–Procesi equation

We consider the problem of the existence of the global solutions and formation of singularities for a b-family of equations which includes the Camassa–Holm and Degasperis–Procesi equation. We also consider the problem of the uniformly continuity of Degasperis–Procesi equation.     

A characteristic description of orthonormal wavelet on subspace of

In this paper, by means of variational iteration method numerical and explicit solutions are computed for some fifth-order Korteweg-de Vries equations, without any linearization or weak nonlinearity assumptions. These equations are the Kawahara equation, Lax’s fifth-order KdV equation and Sawada–Kotera equation. Comparison with Adomian decomposition method reveals that the variational iteration method is easier to be implemented. We conclude that the method is a promising method to various kinds of fifth-order Korteweg-de Vries equations.     

Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force

In this paper, we consider the equations of Magnetohydrodynamics with Coulomb force which is of hyperbolic–parabolic–elliptic mixed type. By constructing the approximate solutions to the modified system with an artificial pressure term added, global existence of finite energy weak solutions is established via the weak convergence method. More careful argument has been paid to overcome the new difficulty arising from the Poisson term of Coulomb force in two dimensions when the adiabatic exponent is close to one. We also investigate the large-time behavior of such weak solutions after discussing the regularity and uniqueness of solutions to the stationary problem.     

On existence and nonexistence of global solutions of Cauchy–Goursat problem for nonlinear wave equations

We consider the Cauchy–Goursat initial characteristic problem for nonlinear wave equations with power nonlinearity. Depending on the power of nonlinearity and the parameter in an equation we investigate the problem on existence and nonexistence of global solutions of the Cauchy–Goursat problem. The question on local solvability of the problem is also considered.     

Investigation and solution of boundary-value problems with parameters by numerical-analytic method

We suggest a modification of the numerical-analytic iteration method. This method is used for studying the problem of existence of solutions and for constructing approximate solutions of nonlinear two-point boundary-value problems for ordinary differential equations with unknown parameters both in the equation and in boundary conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1031–1042, August, 1994.     

Existence of global weak solutions for a phase–field model of a vesicle moving into a viscous incompressible fluid

We consider in this article a model of vesicle moving into a viscous incompressible fluid. The vesicle is described through a phase–field equation and through a transport equation modeling the local incompressibility of its membrane. The equations for the fluid are the classical Navier–Stokes equations with a force resulting from the presence of the vesicle. Our main result states the existence of weak solutions for the corresponding system. The proof is based on compactness/monotonicity arguments. Copyright © 2013 John Wiley & Sons, Ltd.     

Global weak solutions and wave breaking phenomena to the periodic Degasperis–Procesi equation with strong dispersion

Considered herein is the initial-value problem for the periodic Degasperis–Procesi equation with a strong dispersive term that is an approximation to the incompressible Euler equation for shallow water waves. The existence and uniqueness of global weak solutions are established. Moreover, the periodic peakon and shockpeakon solutions of the equation are constructed.     

On the Equivalence of Solutions for a Class of Stochastic Evolution Equations in a Banach Space

We study a class of stochastic evolution equations in a Banach space E driven by cylindrical Wiener process. Three different analytical concepts of solutions: generalised strong, weak and mild are defined and the conditions under which they are equivalent are given. We apply this result to prove existence, uniqueness and continuity of weak solutions to stochastic delay evolution equations. We also consider two examples of these equations in non-reflexive Banach spaces: a stochastic transport equation with delay and a stochastic delay McKendrick equation.     

Construction of weak solutions of a certain stochastic Navier–Stokes equation

We prove the existence of weak solutions of stochastic Navier–Stokes equation on a two-dimensional torus, which appears in a certain variational problem. Our equation does not satisfy the coercivity condition. We construct its weak solutions due to an approximation by a sequence of solutions of equations with enlarged viscosity terms and then by showing an a priori estimate for them.     

Stabilization of the statistical solution of the parabolic equation

We study the convergence of the statistical solutions of the parabolic equation. Under some mixing condition (in the sense of Rosenblatt) for initial measure and natural assumptions on the coefficients of the equation we prove weak convergence to the Gaussian distribution. Similar results for the hyperbolic equations were obtained in [1–4].     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motions and with discontinuous coefficients

We prove an existence theorem for weak solutions of stochastic differential equations with standard and fractional Brownian motions and with discontinuous coefficients. A weak solution of an equation is understood as a weak solution of a stochastic differential inclusion constructed on the basis of the equation. We derive conditions providing the absence of blow-up in weak solutions.     

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data:

We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the L^p–L^q estimates of fundamental solutions for linear damped wave equations [K. Nishihara, L^p–L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The L^p–L^q estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and L^p–L^q estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, L^p–L^q estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method [Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701].     

Operator Hamiltonians and anticanonic equations with periodic coefficients

In a Hilbert space we study Hamiltonians and anticanonic equations with periodic coefficients. We prove existence theorems for the solutions of ill-posed Cauchy problems for the given equations. Following Krein we define the notion of the genus of the spectrum points of the monodromy operator of an equation of the class being studied. We formulate existence and uniqueness theorems for the solutions when determining the reflected and the transmitted waves for a specified incident wave. The theory developed is applied to the study of cylindrical waveguides with a periodic filling.Translated from Problemy Matematicheskogo Analiza. No. 4: Integralnye i Differentsial'nye Operatory. Differentsial'nye Uraveniya, pp. 9–36, 1973.     

Weak solutions to a nonlinear variational wave equation and some related problems

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L ^p Young measure theory and related compactness results, in the first section. Then we use the L ^p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.     

A limit theorem for a class of stochastic integral equations

We obtain an existence and uniqueness theorem for a measurable solution of an Itô-Volterra stochastic integral equation and a limit theorem on convergence of sequences of solutions of such equations.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 47–53, 1987.     

The Vlasov–Poisson–Fokker–Planck equation in an interval with kinetic absorbing boundary conditions

We study the initial–boundary value problem for the Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions. We first prove the existence of weak solutions of the linearized equation in an interval with absorbing boundary conditions. Moreover, the weak solution converges to zero exponentially in time. Then we extend the above results to the fully nonlinear Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions; the existence and the longtime behavior of weak solutions. Finally, we prove that the weak solution is actually a classical solution by showing the hypoellipticity of the solution away from the grazing set and the Hölder continuity of the solution up to the grazing set.