Singular solutions to systems of conservation laws and their algebraic aspects

V. M. Shelkovich*

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Displaying similar documents to “Singular solutions to systems of conservation laws and their algebraic aspects”

On the differentiability of certain saltus functions

Gerald Kuba (2011)

Colloquium Mathematicae

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We investigate several natural questions on the differentiability of certain strictly increasing singular functions. Furthermore, motivated by the observation that for each famous singular function f investigated in the past, f’(ξ) = 0 if f’(ξ) exists and is finite, we show how, for example, an increasing real function g can be constructed so that $g^{'} (x) = 2^{x}$ for all rational numbers x and g’(x) = 0 for almost all irrational numbers x.

Results on generalized models and singular products of distributions in the Colombeau algebra $𝒢 (ℝ)$

Blagovest Damyanov (2015)

Commentationes Mathematicae Universitatis Carolinae

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Models of singularities given by discontinuous functions or distributions by means of generalized functions of Colombeau have proved useful in many problems posed by physical phenomena. In this paper, we introduce in a systematic way generalized functions that model singularities given by distributions with singular point support. Furthermore, we evaluate various products of such generalized models when the results admit associated distributions. The obtained results follow the idea...

Estimates for maximal singular integrals

Loukas Grafakos (2003)

Colloquium Mathematicae

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It is shown that maximal truncations of nonconvolution L²-bounded singular integral operators with kernels satisfying Hörmander’s condition are weak type (1,1) and $L^{p}$ -bounded for 1 < p< ∞. Under stronger smoothness conditions, such estimates can be obtained using a generalization of Cotlar’s inequality. This inequality is not applicable here and the point of this article is to treat the boundedness of such maximal singular integral operators in an alternative way.

Estimates for singular integrals and extrapolation

Shuichi Sato (2009)

Studia Mathematica

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We study singular integrals with rough kernels, which belong to a class of singular Radon transforms. We prove certain estimates for the singular integrals that are useful in an extrapolation argument. As an application, we prove $L^{p}$ boundedness of the singular integrals under a certain sharp size condition on their kernels.

Singular $φ$ -Laplacian third-order BVPs with derivative dependance

Smaïl Djebali, Ouiza Saifi (2016)

Archivum Mathematicum

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This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a $φ$ -Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition...

Young's (in)equality for compact operators

Gabriel Larotonda (2016)

Studia Mathematica

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If a,b are n × n matrices, T. Ando proved that Young’s inequality is valid for their singular values: if p > 1 and 1/p + 1/q = 1, then $λ_{k} (| a b * |) \leq λ_{k} {(1 / p | a |}^{p} + 1 / {q | b |}^{q})$ for all k. Later, this result was extended to the singular values of a pair of compact operators acting on a Hilbert space by J. Erlijman, D. R. Farenick and R. Zeng. In this paper we prove that if a,b are compact operators, then equality holds in Young’s inequality if and only if ${| a |}^{p} = {| b |}^{q}$ .

Positive solutions for a singular second-order ordinary differential equation.

Zhou, W.S., Cai, S.F. (2006)

Lobachevskii Journal of Mathematics

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Disjoint strict singularity of inclusions between rearrangement invariant spaces

Francisco L. Hernández, Víctor M. Sánchez, Evgueni M. Semenov (2001)

Studia Mathematica

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It is studied when inclusions between rearrangement invariant function spaces on the interval [0,∞) are disjointly strictly singular operators. In particular suitable criteria, in terms of the fundamental function, for the inclusions $L ¹ \cap L^{\infty} ↪ E$ and $E ↪ L ¹ + L^{\infty}$ to be disjointly strictly singular are shown. Applications to the classes of Lorentz and Marcinkiewicz spaces are given.

On complete solutions and complete singular solutions of second order ordinary differential equations

Masatomo Takahashi (2007)

Colloquium Mathematicae

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A complete solution of an implicit second order ordinary differential equation is defined by an immersive two-parameter family of geometric solutions on the equation hypersurface. We show that a completely integrable equation is either of Clairaut type or of first order type. Moreover, we define a complete singular solution, an immersive one-parameter family of singular solutions on the contact singular set. We give conditions for existence of a complete solution and a complete singular...

Strictly singular inclusions of rearrangement invariant spaces and Rademacher spaces

Sergei V. Astashkin, Francisco L. Hernández, Evgeni M. Semenov (2009)

Studia Mathematica

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If G is the closure of $L_{\infty}$ in exp L₂, it is proved that the inclusion between rearrangement invariant spaces E ⊂ F is strictly singular if and only if it is disjointly strictly singular and E ⊊ G. For any Marcinkiewicz space M(φ) ⊂ G such that M(φ) is not an interpolation space between $L_{\infty}$ and G it is proved that there exists another Marcinkiewicz space M(ψ) ⊊ M(φ) with the property that the M(ψ) and M(φ) norms are equivalent on the Rademacher subspace. Applications are given and a question...

On nonincreasing singular solutions to the Emden-Fowler equation.

Rabtsevich, V.A. (2000)

Memoirs on Differential Equations and Mathematical Physics

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On nondecreasing singular solutions of the Emden-Fowler equation.

Rabtsevich, V.A. (2000)

Memoirs on Differential Equations and Mathematical Physics

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Singular nonlinear problem for ordinary differential equation of the second order

Irena Rachůnková, Jan Tomeček (2007)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The paper deals with the singular nonlinear problem $u^{''} (t) + f (t, u (t), u^{'} (t)) = 0, u (0) = 0, u^{'} (T) = ψ (u (T)),$ where $f \in 𝐶𝑎𝑟 ((0, T) \times D)$ , $D = (0, \infty) \times$ . We prove the existence of a solution to this problem which is positive on $(0, T]$ under the assumption that the function $f (t, x, y)$ is nonnegative and can have time singularities at $t = 0$ , $t = T$ and space singularity at $x = 0$ . The proof is based on the Schauder fixed point theorem and on the method of a priori estimates.

Behaviour of $L^{r}$ -Dini singular integrals in weighted $L^{1}$ spaces

Osvaldo Capri, Carlos Segovia (1989)

Studia Mathematica

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Singular perturbations of ordinary differential equations in Colombeau spaces

Marko Nedeljkov, Danijela Rajter (2000)

Publications de l'Institut Mathématique

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Boundedness of vector-valuedB-singular integral operators in Lebesgue spaces

Seyda Keles, Mehriban N. Omarova (2017)

Open Mathematics

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We study the vector-valued B-singular integral operators associated with the Laplace-Bessel differential operator △B=∑k=1n−1∂ 2∂x k 2+(∂2∂x n 2+2vxn∂∂x n),v>0. $▵_{B} = \sum_{k = 1}^{n - 1} \frac{\partial^{2}}{\partial x_{k}^{2}} + (\frac{\partial^{2}}{\partial x_{n}^{2}} + \frac{2 v}{x_{n}} \frac{\partial}{\partial x_{n}}), v > 0 .$ We prove the boundedness of vector-valued B-singular integral operators A from [...] Lp,v(R+n,H1)toLp,v(R+n,H2), $L_{p, v} (ℝ_{+}^{n}, H_{1}) to L_{p, v} (ℝ_{+}^{n}, H_{2}),$ 1 < p < ∞, where H1 and H2 are separable Hilbert spaces.