Search

In this paper we study smoothness properties of solutions to a two-dimensional Kawahara equation. We show that the equation's dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data ϕ possesses certain regularity and sufficient decay as x → ∞ , then the solution u (t) will be smoother than ϕ for 0 < t ≤ T where T is the existence time of the solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Newtonian limit of the relativistic Euler-Poisson equation. Under conditions of the free boundary and vacuum, we prove the existence and uniqueness of local smooth solutions, which converge to the solutions of the classical Euler-Poisson equation at the rate of c − 2 , where c is the speed of light. From the mathematical standpoint, we successfully overcome the strong nonlinearity caused by the Lorentz factor, the vacuum occurring on the moving boundary and the singularity at the center point by applying the weighted Sobolev space, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study several problems on the weakly dissipative periodic Camassa-Holm type equation with quadratic and cubic non-linearities. First, in the periodic setting, the local well-posedness of solutions in the Sobolev space is established via Kato's theory. Then, we establish a relation between the behavior of u x and wave breaking phenomena of solutions, and give a sufficient condition on the initial data to guarantee the occurrence of wave breaking. Finally, we obtain the global existence of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the linear and semilinear Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the linear MGT model, by utilizing WKB analysis associated with Fourier analysis, we derive some L 2 estimates of solutions, which improve those in the previous research [51]. Furthermore, asymptotic profiles of the solution and an approximate relation in a framework of the weighted L 1 space are derived. Next, with the aid of the classical energy method and Hardy's inequality, we get singular limit results for an energy and the solution itself. Concerning the semilinear MGT model, basing on the obtained sharp L 2 estimates and constructing time-weighted Sobolev spaces, we investigate global (in time) existence of Sobolev solutions with different regularities. Finally, under a sign assumption on initial data, nonexistence of global (in time) weak solutions is proved by applying a test function method. [ABSTRACT FROM AUTHOR]

In this paper, we establish the well-posedness for the third grade fluid equation perturbed by a multiplicative white noise. This equation describes the motion of a non-Newtonian fluid of differential type with relevant viscoelastic properties. We are faced with a strongly nonlinear stochastic partial differential equation supplemented with a Navier slip boundary condition. Taking the initial condition in the Sobolev space H 2 , we show the existence and the uniqueness of the strong (in the probability sense) solution in a two dimensional and non axisymmetric bounded domain. [ABSTRACT FROM AUTHOR]

In this paper, we consider the long wavelength limit for the Euler-Poisson system arising in plasma including three species. It is demonstrated that when the plasma has critical densities, the modified Korteweg-de Vries (mKdV) equation can be derived, under the classical Gardner-Morikawa transform ε 1 / 2 (x − V t) → x , ε 3 / 2 t → t as ε → 0 , with the velocity V depending on the densities. We estimate the error between the mKdV equation and the Euler-Poisson system in Sobolev spaces. The mKdV dynamics can be seen at time interval of order O (ε − 1). [ABSTRACT FROM AUTHOR]

In this paper, we apply ∂ ‾ steepest descent method to study the Cauchy problem for the nonlinear short-pulse equation u x t = u + 1 6 (u 3) x x , u (x , 0) = u 0 (x) ∈ H (R) , where H (R) = W 3 , 1 (R) ∩ H 2 , 2 (R) is a weighted Sobolev space. The solution of the short-pulse equation is constructed via a solution of Riemann-Hilbert problem in the new scale (y , t). In any fixed space-time cone C (y 1 , y 2 , v 1 , v 2) = { (y , t) ∈ R 2 : y = y 0 + v t , y 0 ∈ [ y 1 , y 2 ] , v ∈ [ v 1 , v 2 ] } , we compute the long time asymptotic expansion of the solution u (x , t) , which implies soliton resolution conjecture consisting of three terms: the leading order term can be characterized with an N (I) -soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone; the second | t | − 1 / 2 order term coming from soliton-radiation interactions on continuous spectrum up to an residual error order O (| t | − 1) from a ∂ ‾ equation. Our results also show that soliton solutions of the short-pulse equation are asymptotically stable. [ABSTRACT FROM AUTHOR]

In this paper we study the initial-value problem associated with the dispersion generalized-Benjamin-Ono-Zakharov-Kuznetsov equation, u t + D x a + 1 ∂ x u + u x y y + u u x = 0 , a ∈ (0 , 1). More specifically, we study the persistence property of the solution in the weighted anisotropic Sobolev spaces H (1 + a) s , 2 s (R 2) ∩ L 2 ((x 2 r 1 + y 2 r 2 ) d x d y) , for appropriate s , r 1 and r 2. By establishing unique continuation properties we also show that our results are sharp with respect to the decay in the x -direction. [ABSTRACT FROM AUTHOR]

For s < 3 / 2 , it is known (see [24]) that the Cauchy problem for the b-family of equations is ill-posed in Sobolev spaces H s when b > 1. The proof of ill-posedness depends naturally on the value of b , and is based on the construction of peakon–antipeakon solutions with interesting properties which allows to make conclusion on ill-posedness. In this context the construction of such type of the solution for b < 1 is very attractive problem which help shed light on the ill-posedness problem in this case. Thus, in the present paper we consider the ill-posedness of the b-family of equations with additional term for insufficiently investigated case b < 1 on the line. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the Cauchy problem of the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation, which has the dissipative term − ∂ x 2 u. This is known that the 2D Zakharov-Kuznetsov equation is well-posed in H s (R 2) for s > 1 / 2 , and the 2D nonlinear parabolic equation with quadratic derivative nonlinearity is well-posed in H s (R 2) for s ≥ 0. By using the Fourier restriction norm with dissipative effect, we prove the well-posedness for ZKB equation in H s (R 2) for s > − 1 / 2. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the smoothness properties of solutions to the KP-I equation. We show that the equation''s dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data ϕ possesses certain regularity and sufficient decay as , then the solution will be smoother than ϕ for where T is the existence time of the solution. [Copyright &y& Elsevier]

Abstract: In this paper, we study the Stokes system in the half-space , with . We give existence and uniqueness results in weighted Sobolev spaces. After the central case of the generalized solutions, we are interested in strong solutions and symmetrically in very weak solutions by means of a duality argument. [Copyright &y& Elsevier]

Abstract: Let be the solution of the heat equation on subject to on . The main goal of this paper is to characterize such a nonnegative measure on that induces a bounded embedding from the Sobolev space , into the Lebesgue space , . [Copyright &y& Elsevier]

In this paper, we answer affirmatively an open problem (cf. Theorem 4′ in Ferrero and Gazzola (J. Differential Equations 177 (2001) 494): Let Ω∋0 be an open-bounded domain, Ω⊂RN(N⩾5) and assume that 0⩽μ<(N-2/2)2-(N+2/N)2, then, for all λ>0 there exists a nontrivial solution with critical level in the range (0,1/NSμN/2) for the problem -Δu-μ u/|x|2=λu+|u|2*-2u in Ω; u=0 on ∂Ω. [Copyright &y& Elsevier]