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.     

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.     

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.     

The global attractor of the viscous Fornberg–Whitham equation

This paper aims to present a proof of the existence of the attractor for the one-dimensional viscous Fornberg–Whitham equation. In this paper, the global existence of solution to the viscous Fornberg–Whitham equation in L² under the periodic boundary conditions is studied. By using the time estimate of the Fornberg–Whitham equation, we get the compact and bounded absorbing set and the existence of the global attractor for the viscous Fornberg–Whitham equation.     

Initial–boundary value problems for conservation laws with source terms and the Degasperis–Procesi equation

We consider conservation laws with source terms in a bounded domain with Dirichlet boundary conditions. We first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition. Equipped with this formulation, we go on to establish the well-posedness of entropy solutions to the initial–boundary value problem. The proof utilizes the kinetic formulation and the averaging lemma. Finally, we make use of these results to demonstrate the well-posedness in a class of discontinuous solutions to the initial–boundary value problem for the Degasperis–Procesi shallow water equation, which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term.     

Integrable Hierarchy, 3×3 Constrained Systems, and Parametric Solutions

This paper provides a new integrable hierarchy. The DP equation: m _t +um _x +3mu _x =0, m=u–u _xx , proposed recently by Degasperis and Procesi, is the first member in the negative order hierarchy while the first equation in the positive order hierarchy is: m _t =4(m ^–2/3) _x –5(m ^–2/3) _xxx +(m ^–2/3) _xxxxx . The whole hierarchy is shown Lax-integrable through solving a key matrix equation. To obtain the parametric solutions for the whole hierarchy, we separately discuss the negative order and the positive order hierarchies. For the negative order hierarchy, its 3×3 Lax pairs and corresponding adjoint representations are cast in Liouville-integrable Hamiltonian canonical systems under the Dirac–Poisson bracket defined on a symplectic submanifold of R ^6N . Based on the integrability of those finite-dimensional canonical Hamiltonian systems we give the parametric solutions of all equations in the negative order hierarchy. In particular, we obtain the parametric solution of the DP equation. Moreover, for the positive order hierarchy, we consider a different constraint and process a procedure similar to the negative case to obtain the parametric solutions of the positive order hierarchy. In a special case, we give the parametric solution of the 5th-order PDE m _t =4(m ^–2/3) _x –5(m ^–2/3) _xxx +(m ^–2/3) _xxxxx . Finally, we discuss the stationary solutions of the 5th-order PDE, which may be included in the parametric solution.     

An a Posteriori Error Estimate for a Semi-Lagrangian Scheme for Hamilton–Jacobi Equations

We present an a posteriori estimate for a first order semi-Lagrangian method for Hamilton–Jacobi equations. The result requires piecewise C ^1,1 regularity of the viscosity solution and is stated for the Bellman equation related to the infinite horizon problem, although it can be applied to more general Hamilton–Jacobi equations with convex Hamiltonians. This estimate suggests different numerical indicators that can be used to construct an adaptive algorithm for the approximation of the viscosity solution.     

The existence of strong solution for a class of fully nonlinear equation

This article mainly investigates the existence of global strong solution of a class of fully nonlinear evolution equation and the strong solution of its steady-state equation. By using the T-compulsorily weakly continuous operator theory, the existence of the global strong solution of the fully nonlinear evolution equation is obtained. In addition, based on the acute angle principle, the W^2,p-strong solution for the corresponding stationary equation is also derived.     

Attractor for the viscous two-component Camassa-Holm equation

This paper is concerned with the attractor for a viscous two-component generalization of the Camassa-Holm equation subject to an external force, where the viscosity term is given by a second order differential operator. The global existence of solution to the viscous two-component Camassa-Holm equation with the periodic boundary condition is studied. We obtain the compact and bounded absorbing set and the existence of the global attractor in H²×H² for the viscous two-component Camassa-Holm equation by uniform prior estimate and many inequalities.     

On second grade fluids with vanishing viscosity

We consider in ℝⁿ (n = 2, 3) the equation of a second grade fluid with vanishing viscosity, also known as Camassa-Holm equation. We prove local existence and uniqueness of solutions for smooth initial data. We also give a blow-up condition which implies that the solution is global for n = 2. Finally, we prove the convergence of the solutions of second grade fluid equation to the solution of the Camassa-Holm equation as the viscosity tends to zero.     

A generalized solution in W2 1(0,1) of a problem concerning the deflection of a beam

A new method is proposed for formulating a boundary-value problem for a fourth-order ordinary differential equation with a solution in W₂ ¹(0, 1). This generalized formulation is based on a system of second-order equations with coefficients in W₂ ^–1 (0, 1). The existence and uniqueness of the indicated solution in this class is proven.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 90–96, 1989.     

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].     

Conservative finite difference schemes for the Degasperis–Procesi equation

We consider the numerical integration of the Degasperis–Procesi equation, which was recently introduced as a completely integrable shallow water equation. For the equation, we propose nonlinear and linear finite difference schemes that preserve two invariants associated with the bi-Hamiltonian form of the equation at the same time. We also prove the unique solvability of the schemes, and show some numerical examples.     

Modified Camassa–Holm and Degasperis–Procesi Equations Solved by Adomian’s Decomposition Method and Comparison with HPM and Exact Solutions

Many physical and scientific phenomena are modeled by nonlinear partial differential equations (NPDEs); it is difficult to handle nonlinear part of these equations. Recently some analytical methods are applied to solve such equations. In this work, modified Camassa–Holm and Degasperis–Procesi equation is studied. Adomian’s decomposition method (ADM) is applied to obtain solution of this equation. The results are compared to those of homotopy perturbation method (HPM) and exact solution. The study highlights the significant features of the employed method and its ability to handle nonlinear partial differential equations.     

The global attractor of g-Navier–Stokes equations with linear dampness on

In this paper, the long time behaviors of g-Navier–Stokes equations with linear dampness on R² were investigated. By using the energy equation method, the existence of the global attractor for the equations was proved without the restriction of the forcing term belonging to some weighted Sobolev space. Moreover, the estimation of the Hausdorff and Fractal dimensions of such attractors were also obtained.     

The Cauchy problem for a class of quasilinear parabolic equations

The present paper is concerned with the Cauchy problem for the parabolic equation u_t+H(t,x,u,u)=u. New conditions guaranteeing the global classical solvability are formulated. Moreover, it is shown that the same conditions guarantee the global existence of the Lipschitz continuous viscosity solution for the related Hamilton–Jacobi equation. Mathematics Subject Classification (2000) 35K15, 35F25     

Nonhomogeneous Neumann problem for the Poisson equation in domains with an external cusp

In this work we analyze the existence and regularity of the solution of a nonhomogeneous Neumann problem for the Poisson equation in a plane domain Ω with an external cusp. In order to prove that there exists a unique solution in H¹(Ω) using the Lax–Milgram theorem we need to apply a trace theorem. Since Ω is not a Lipschitz domain, the standard trace theorem for H¹(Ω) does not apply, in fact the restriction of H¹(Ω) functions is not necessarily in L²(∂Ω). So, we introduce a trace theorem by using weighted Sobolev norms in Ω. Under appropriate assumptions we prove that the solution of our problem is in H²(Ω) and we obtain an a priori estimate for the second derivatives of the solution.     

Solvability of the mixed initial boundary-value problem for a parabolic equation with a nonlocal condition of first kind

The spectral method is applied to solve the mixed initial boundary-value problem for a parabolic equation with nonhomogeneous boundary conditions, one of which is nonlocal. We prove existence and uniqueness of the generalized solution of this problem in the Sobolev class W ₂ ^1,0 and represent it as a biorthogonal series. We also consider optimal control by the right-hand side of the equation, which is constructed as a biorthogonal series in the root functions of the spectral problem.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 209–220, 2002.     

Smooth Solution of the Compressible Navier–Stokes Equations in an Unbounded Domain with Inflow Boundary Condition

The barotropic compressible Navier–Stokes equations in an unbounded domain are studied. We prove the unique existence of the solution (u, p) of the system (1.1) in the Sobolev spaceH^{k + 3} × H^{k + 2}provided that the derivatives of the data of the problem are sufficiently small, wherek ≥ 0 is any integer. The proof follows from an analysis of the linearized problem, the solvability of the continuity equation, and the Schauder fixed point theory. Similar smoothness results are obtained for a linearized form of (1.1).     

The Semilinear Wave Equation on Asymptotically Euclidean Manifolds

We consider the quadratically semilinear wave equation on (? ^d , 𝔤), d ≥ 3. The metric 𝔤 is non-trapping and approaches the Euclidean metric like ?x?^?ρ. Using Mourre estimates and the Kato theory of smoothness, we obtain, for ρ > 0, a Keel–Smith–Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for ρ ≥ 1. Long time existence means that, for all n > 0, the life time of the solution is a least δ^?n , where δ is the size of the initial data in some appropriate Sobolev space. Moreover, for d ≥ 4 and ρ > 1, we obtain global existence for small data.