Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws

In this paper we provide a generalized version of the Glimm scheme to establish the global existence of weak solutions to the initial-boundary value problem of 2×2 hyperbolic systems of conservation laws with source terms. We extend the methods in [J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. Dissertation, Stanford University, 1982; J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem, J. Differential Equations 222 (2006) 515-549] to construct the approximate solutions of Riemann and boundary Riemann problems, which can be adopted as the building block of approximate solutions for our initial-boundary value problem. By extending the results in [J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697-715] and showing the weak convergence of residuals, we obtain stability and consistency of the scheme.     

Existence, non-existence and asymptotic behavior of global solutions to the Cauchy problem for systems of semilinear hyperbolic equations with damping terms

We consider the Cauchy problem for systems of semilinear hyperbolic equations. Using the L_p→L_q type estimation for the corresponding linear parts, the existence and uniqueness of weak global solutions are investigated. We also established the behavior of solutions and their derivatives as t→+∞. Using the method of test functions developed in the works (Mitidieri and Pokhozhaev, 2001 [11], Veron and Pohozaev, 2001 [12] and Caristi, 2000 [23]) we obtain the analogue of the Fujita-Hayakawa type criterion for the absence of global solutions to some system of semilinear hyperbolic inequalities with damping. It follows that the conditions of existence theorem imposed on the growth of nonlinear parts are exact in some sense.     

On recent developments in the theory of abstract differential equations with fractional derivatives

This note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives. We show that the existence results in [Agarwal et al. (2009) [1], Belmekki and Benchohra (2010) [2], Darwish et al. (2009) [3], Hu et al. (2009) [4], Mophou and N’Guérékata (2009) [6] and [7], Mophou (2010) [8] and [9], Muslim (2009) [10], Pandey et al. (2009) [11], Rashid and El-Qaderi (2009) [12] and Tai and Wang (2009) [13]] are incorrect since the considered variation of constant formulas is not appropriate. In this note, we also consider a different approach to treat a general class of abstract fractional differential equations.     

On global existence and attractivity results for nonlinear functional integral equations

In this paper two existence results concerning the global attractivity and global asymptotic attractivity for a certain functional nonlinear integral equation are proved. Our existence results include several existence as well as attractivity results obtained earlier by Banas and Dhage (2008) [1], Hu and Yan (2006) [3], Dhage (2009) [15] and Banas and Rzepka (2003) [7] as special cases under some weaker Lipschitz conditions. A measure theoretic fixed point theorem of Dhage (2008) [6] is used in formulating our main results and the characterizations of solutions are obtained in the space of functions defined, continuous and bounded on unbounded intervals.     

Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum

This paper is concerned with existence of global weak solutions to a class of compressible Navier-Stokes equations with density-dependent viscosity and vacuum. When the viscosity coefficient μ is proportional to ρθ with , a global existence result is obtained which improves the previous results in Fang and Zhang (2004) [4], Vong et al. (2003) [27], Yang and Zhu (2002) [30]. Here ρ is the density. Moreover, we prove that the domain, where fluid is located on, expands outwards into vacuum at an algebraic rate as the time grows up due to the dispersion effect of total pressure. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., θ=1, γ=2).     

A new proof of the existence of weak solutions to a model for phase evolution driven by material forces

We prove the existence of weak solutions to a one‐dimensional initial‐boundary value problem for a model system of partial differential equations, which consists of a sub‐system of linear elasticity and a nonlinear non‐uniformly parabolic equation of second order. To simplify the existence proof of weak solutions in the 2006 paper of Alber and Zhu, we replace the function in that work by . The model is formulated by using a sharp interface model for phase transformations that are driven by material forces. Copyright © 2017 John Wiley & Sons, Ltd.     

A regularity result for quasilinear parabolic systems

We consider bounded, weak solutions of certain quasilinear parabolic systems of second order. If the solution fulfills a suitable smallness condition, we show that it is H?lder continuous and satisfies an a priori estimate. This is a well known result of Giaquinta and Struwe [3]. Their argument employs the use of Green’s functions, which is completely avoided in our proof. Instead, our crucial tool is a weak Harnack inequality for supersolutions due to Trudinger [7] in connection with a technique developed by L.Caffarelli [1]. Received: 25 September 2006     

Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations

This paper is concerned with classical solutions to the interaction of two arbitrary planar rarefaction waves for the self-similar Euler equations in two space dimensions. We develop the direct approach, started in Chen and Zheng (in press) [3], to the problem to recover all the properties of the solutions obtained via the hodograph transformation of Li and Zheng (2009) [14]. The direct approach, as opposed to the hodograph transformation, is straightforward and avoids the common difficulties of the hodograph transformation associated with simple waves and boundaries. The approach is made up of various characteristic decompositions of the self-similar Euler equations for the speed of sound and inclination angles of characteristics.     

Stability for semilinear elliptic variational inequalities depending on the gradient

In this work we give a result concerning the continuous dependence on the data for weak solutions of a class of semilinear elliptic variational inequalities (P_n) with a nonlinear term depending on the gradient of the solution. This paper can be seen as the second part of the work Matzeu and Servadei (2010) [9], in the sense that here we give a stability result for the C^1,α-weak solutions of problem (P_n) found in Matzeu and Servadei (2010) [9] through variational techniques. To be precise, we show that the solutions of (P_n), found with the arguments of Matzeu and Servadei (2010) [9], converge to a solution of the limiting problem (P), under suitable convergence assumptions on the data.     

The lifespan for 3D quasilinear wave equations with nonlinear damping terms

In this paper, we study the initial-boundary value problem for a system of nonlinear wave equations, involving nonlinear damping terms, in a bounded domain Ω. The nonexistence of global solutions is discussed under some conditions on the given parameters. Estimates on the lifespan of solutions are also given. Our results extend and generalize the recent results in [K. Agre, M.A. Rammaha, System of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006) 1235-1270], especially, the blow-up of weak solutions in the case of non-negative energy.     

A robin-neumann problem in bounded domains with splitting boundary

We consider a mixed boundary-value problem for the homogeneous Laplace equation in a bounded domain which boundary splits up into two disjoint smooth components. On the one boundary component we pose a homogeneous Robin condition and an inhomogeneous Neumann condition on the other. We give a weak formulation, interpret this problem as a generalized spectral (eigenvalue) problem in the sense of F.Stummel (cf.[12]) and investigate existence, uniqueness and regularity of weak solutions. This problem is a cut-off version of a basic problem in water-wave theory (cf.Ramm [8], pp.394-395, Simon/Ursell [10] Stoker [11])     

Free boundary value problem for a viscous two-phase model with mass-dependent viscosity

In this paper, we study a free boundary value problem for two-phase liquid-gas model with mass-dependent viscosity coefficient when both the initial liquid and gas masses connect to vacuum with a discontinuity. This is an extension of the paper [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV]. Just as in [S. Evje, K.H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, http://www.irisresearch.no/docsent/emp.nsf/wvAnsatte/SEV], the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid. We give the proof of the global existence and uniqueness of weak solutions when β∈(0,1], which have improved the previous result of Evje and Karlsen, and get the asymptotic behavior result, also we obtain the regularity of the solutions by energy method.     

The degree of an eight-dimensional real quadratic division algebra is 1, 3, or 5

A celebrated theorem of Hopf (1940) [11], Bott and Milnor (1958) [1], and Kervaire (1958) [12] states that every finite-dimensional real division algebra has dimension 1, 2, 4, or 8. While the real division algebras of dimension 1 or 2 and the real quadratic division algebras of dimension 4 have been classified (Dieterich (2005) [6], Dieterich (1998) [3], Dieterich and Öhman (2002) [9]), the problem of classifying all 8-dimensional real quadratic division algebras is still open. We contribute to a solution of that problem by proving that every 8-dimensional real quadratic division algebra has degree 1, 3, or 5. This statement is sharp. It was conjectured in Dieterich et al. (2006) [7].     

Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone

In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257-289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large.     

An Extended Beam Theory for Smart Materials Applications Part II: Static Formation Problems

In this paper we further develop the theory of the extended Timoshenko beam model, as first introduced in Part I [5] of this work, with particular emphasis on applications of the model in formation theory [11], [12]. We begin with formal development of the equilibrium equations of static formation theory in the context of the extended Timoshenko model, giving a rigorous discussion of existence, uniqueness, and regularity of weak solutions with appropriate assumptions on the coefficients. We continue to obtain the fundamental duality relationship in the context of weak solutions and indicate its usefulness in investigations of approximate formability. Optimal formation problems and corresponding necessary conditions for optimality are discussed. We conclude with a discussion of a particular problem of joint optimization of controls and actuator densities in the context of a prismatic extended Timoshenko beam and we present the results of some computational studies. Accepted 14 November 1996     

Existence of multidimensional shock waves and phase boundaries

In this paper, a kind of Riemann problem for the Euler equations in a van der Waals fluid is considered. We constructed the weak solution in multidimensional space which contains one shock front and one subsonic phase boundary. We mainly follow the arguments of Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95; A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1-93] and Métivier's [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479] work. The linear stability results are based on Majda's [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1-95] work for the single shock front and Wang and Xin's [Y.-G. Wang, Z. Xin, Stability and existence of multidimensional subsonic phase transitions, Acta Math. Appl. Sin. 19 (2003) 529-558] work for the single phase boundary. The initial boundary value problem concerned in this paper is different from the boundary value problem for double shock fronts concerned in [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace, Trans. Amer. Math. Soc. 296 (1986) 431-479], we slightly modified Métivier's frame work to establish the existence for the solution to the nonlinear problem.     

Existence of weak solutions for steady viscous fluids with general slip boundary conditions

We study the existence of weak solutions for stationary viscous fluids with general slip boundary conditions in this paper. Applying monotone operator theory, we first establish the existence result of weak solutions for an approximation problem. Then using the compactness methods and the point-wise convergence property of velocity gradients, we get the desired results.     

Regularity for the porous medium equation with variable exponent: The singular case

We extend to the singular case the results of [E. Henriques, J.M. Urbano, Intrinsic scaling for PDEs with an exponential nonlinearity, Indiana Univ. Math. J. 55 (5) (2006) 1701-1721] concerning the regularity of weak solutions of the porous medium equation with variable exponent. The method of intrinsic scaling is used to show that local weak solutions are locally continuous.     

Blow-up theorem for semilinear wave equations with non-zero initial position

One of the features of solutions of semilinear wave equations can be found in blow-up results for non-compactly supported data. In spite of finite propagation speed of the linear wave, we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak. This was first observed by Asakura (1986) [2] finding out a critical decay to ensure the global existence of the solution. But the blow-up result is available only for zero initial position having positive speed.In this paper the blow-up theorem for non-zero initial position by Uesaka (2009) [22] is extended to higher-dimensional case. And the assumption on the nonlinear term is relaxed to include an example, |u|^p−1u. Moreover the critical decay of the initial position is clarified by example.     

Multiple Positive Radial Solutions of Elliptic Equations in an Exterior Domain

Several existence theorems on multiple positive radial solutions of the elliptic boundary value problem in an exterior domain are obtained by using the fixed point index theory. Our conclusions are essential improvements of the results in [7], [10] and [13].