Weak Solutions Of Equation Research Articles

On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis

Abstract Given the abstract evolution equation y ′ ( t ) = A y ( t ) , t ∈ ℝ , y^{\prime} (t)=Ay(t),t\in {\mathbb{R}}, with a scalar type spectral operator A in a complex Banach space, we find conditions on A, formulated exclusively in terms of the location of its spectrum in the complex plane, necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1 \beta \ge 1 , in particular analytic or entire, on ℝ {\mathbb{R}} . We also reveal certain inherent smoothness improvement effects and show that if all weak solutions of the equation are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded. The important particular case of the equation with a normal operator A in a complex Hilbert space follows immediately.

On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

Abstract Given the abstract evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array}$$ with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire, on the open semi-axis (0, ∞). Also, revealed is a certain interesting inherent smoothness improvement effect.

On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis

Given the abstract evolution equation y′(t)=Ay(t), t∈R, with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on R. The important case of the equation with a normal operator A in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on R.

Energy Conservation for the Weak Solutions of the Compressible Navier–Stokes Equations

In this paper, we prove the energy conservation for the weak solutions of the compressible Navier-Stokes equations for any time $t>0$, under certain conditions. The results hold for the renormalized solutions of the equations with constant viscosities, as well as the weak solutions of the equations with degenerate viscosity. Our conditions do not depend on the dimensions. The energy may conserve on the vacuum for the compressible Navier-Stokes equations with constant viscosities. Our results are the first ones on the energy conservation for the weak solutions of the compressible Navier-Stokes equations.

Variational Solutions of the Pricing PIDEs for European Options in Lévy Models

One of the fundamental problems in financial mathematics is to develop efficient algorithms for pricing options in advanced models such as those driven by Lévy processes. Essentially there are three approaches in use. These are Monte Carlo, Fourier transform and partial integro-differential equation (PIDE)-based methods. We focus our attention here on the latter. There is a large arsenal of numerical methods for efficiently solving parabolic equations that arise in this context. Especially Galerkin and Galerkin-inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense.The contribution of this paper is therefore to analyse weak solutions of the Kolmogorov backward equations which are related to prices of European options in (time-inhomogeneous) Lévy models and to establish a precise link between the prices and the weak solutions of these equations. The resulting relation is a Feynman–Kac representation of the solution as a conditional expectation. Our special concern is to provide a framework that is able to cover both, the common types of European options and a wide range of advanced models in which these derivatives are priced.An application to financial models requires in particular to admit pure jump processes such as generalized hyperbolic processes as well as unbounded domains of the equation. In order to deal at the same time with the typical pay-offs that can arise, the weak formulation of the equation is based on exponentially weighted Sobolev–Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options.

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.

On the Differentiabilityof Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator

For the evolution equation y′(t) = Ay(t) with a scalar type spectral operator A in a Banach space, conditions on A are found that are necessary and sufficient for all weak solutions of the equation on [0, ∞) to be strongly infinite differentiable on [0, ∞) or [0, ∞). Certain effects of smoothness improvement of the weak solutions are analyzed.

On a Problem Connected with Navier-stokes Equations in Non Cylindrical Domains

In this study we showed the existence of weak solutions of equations that represent flows of a non-homogeneous viscous incompressible fluids in a non cylindrical domain in R3 . The classical Navier-stokes equation is a particular case of the equations here considered.

Stability of Solution of a Class of Quasilinear Elliptic Equations

In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equations $$ - {\text{div}}\;A_{p} {\left( {x,u} \right)} + a_{p} {\left( {x,u} \right)} = f{\left( x \right)} $$ in space W (θ, p)(Ω), which is bigger than W (1, p)(Ω). Next, using revise reverse Holder inequality we prove that if ω c is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈ U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.

HARNACK ESTIMATES FOR WEAK SOLUTIONS OF EQUATIONS OF NON-NEWTONIAN POLYTROPIC FILTRATION

An intrinsic Harnack estimate and some sup-estimates are established for nonnegative weak solutions of equations of non-Newtonian polytropic filtration ut-div(|Dum|p-2Dum)=0, m(p-1) 0, p>1.

On stationary flow of asymmetric fluids with diffusion

We prove existence and uniqueness theorems for weak solutions of equations describing stationary isothermic motion of a mixture of two viscous incompressible fluids with asymmetric stress tensor, in a bounded subset of ℝ3. The model of the flow we consider here assumes that some of coefficients characterizing isotropic properties of the fluid equal zero.

Derivatives of continuous weak solutions of linear elliptic equations

Let L be a linear elliptic differential operator in n-space of order m. Let M denote the adjoint operator to L. An integrable function u is called a weak solution of the equation L[u] = f in a domain D, if for every w of class C ∞, which vanishes outside a compact subset of D, the relation $$ \int_D {(uM[w]) - fw)d{x_1}...d{x_n}} $$ holds; u will be called a strict solution, if u is of class C m and satisfies L[u] =f in the ordinary sense. One of the remarkable facts concerning elliptic equations is that under suitable regularity assumptions on f and the coefficients of L a weak solution can be differentiated any number of times and is a strict solution. In a recent paper F. E. Browder1 gives the theorem: a weak solution which is square integrable on every compact subset of D is almost everywhere equal to a strict solution, provided f is in C 1 and the coefficients of the j-th derivatives in the operator L are in C m+i . Browder’s proof of this generalization of “Weyl’s lemma” makes use of the fundamental solution of elliptic equations with analytic coefficients.2 Results contained in a paper by K. O. Friedrichs,3 which appears in the same issue, imply the theorem that a weak solution is a strict solution, provided f is in C (n+1)/2 and the coefficients of L are in C m/2.