Search results for "equation"

On the interior regularity of weak solutions to the 2-D incompressible Euler equations

2016

We study whether some of the non-physical properties observed for weak solutions of the incompressible Euler equations can be ruled out by studying the vorticity formulation. Our main contribution is in developing an interior regularity method in the spirit of De Giorgi–Nash–Moser, showing that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity result $$\begin{aligned} u \in L_\mathrm{loc}^{2+\varepsilon }(\Omega _T) \implies \mathrm{local\ regularity} \end{aligned}$$ for weak solutions in the energy space $$L_t^\infty L_x^2$$ , satisfying appropriate vorticity estimates. We also obtain impr…

New degeneration of Fay's identity and its application to integrable systems

2011

In this paper, we find a new degenerated version of Fay's trisecant identity; this degeneration corresponds to the limit when the four points entering the trisecant identity coincide pairwise. This degenerated version of Fay's identity is used to construct algebro-geometric solutions to the multi-component nonlinear Schrodinger equation. This identity also leads to an independent derivation of algebro-geometric solutions to the Davey–Stewartson equations previously obtained in [17] in the framework of the Krichever scheme. We also give the condition of smoothness of the obtained solutions.

Large-x Analysis of an Operator-Valued Riemann–Hilbert Problem

2015

International audience; The purpose of this paper is to push forward the theory of operator-valued Riemann-Hilbert problems and demonstrate their effectiveness in respect to the implementation of a non-linear steepest descent method a la Deift-Zhou. In this paper, we demonstrate that the operator-valued Riemann-Hilbert problem arising in the characterization of so-called c-shifted integrable integral operators allows one to extract the large-x asymptotics of the Fredholm determinant associated with such operators.

Nonlinear scalar field equations with general nonlinearity

2018

Consider the nonlinear scalar field equation \begin{equation} \label{a1} -\Delta{u}= f(u)\quad\text{in}~\mathbb{R}^N,\qquad u\in H^1(\mathbb{R}^N), \end{equation} where $N\geq3$ and $f$ satisfies the general Berestycki-Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski [30] made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski's results on the scalar field equation. T…

Linearization of complex hyperbolic Dulac germs

2021

We prove that a hyperbolic Dulac germ with complex coefficients in its expansion is linearizable on a standard quadratic domain and that the linearizing coordinate is again a complex Dulac germ. The proof uses results about normal forms of hyperbolic transseries from another work of the authors.

A note on the unirationality of a moduli space of double covers

2010

In this note we look at the moduli space $\cR_{3,2}$ of double covers of genus three curves, branched along 4 distinct points. This space was studied by Bardelli, Ciliberto and Verra. It admits a dominating morphism $\cR_{3,2} \to {\mathcal A}_4$ to Siegel space. We show that there is a birational model of $\cR_{3,2}$ as a group quotient of a product of two Grassmannian varieties. This gives a proof of the unirationality of $\cR_{3,2}$ and hence a new proof for the unirationality of ${\mathcal A}_4$.

Regular solutions of transmission and interaction problems for wave equations

1989

Consider n bounded domains Ω ⊆ ℝ and elliptic formally symmetric differential operators A1 of second order on Ωi Choose any closed subspace V in , and extend (Ai)i=1,…,n by Friedrich's theorem to a self-adjoint operator A with D(A1/2) = V (interaction operator). We give asymptotic estimates for the eigenvalues of A and consider wave equations with interaction. With this concept, we solve a large class of problems including interface problems and transmission problems on ramified spaces.25,32 We also treat non-linear interaction, using a theorem of Minty29.

Removability theorems for solutions of degenerate elliptic partial differential equations

1993

A group analysis via weak equivalence transformations for a model of tumor encapsulation

2004

A symmetry reduction of a PDEs system, describing the expansive growth of a benign tumour, is obtained via a group analysis approach. The presence in the model of three arbitrary functions suggests the use of Lie symmetries by using the weak equivalence transformations. An invariant classification is given which allows us to reduce the initial PDEs system to an ODEs system. Numerical simulations show a realistic enough description of the physical process.

Unifying vectors and matrices of different dimensions through nonlinear embeddings

2020

Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous parameter $\kappa \in \mathbb{R}$ is being varied, thus allowing the unification of vectors, matrices and tensors in single mathematical structures. This technique is applied to construct warped models in the passage from supergravity in 10 or 11-dimensional spacetimes to 4-dimensional ones. We also show how nonlinear embeddings can be used to connect cellular automata (CAs) to coupled map lattices (CMLs) and to nonlinear partial differential equations, derivi…