Well-posedness Of Initial Value Problem Research Articles

A viscoelastic flow model of Maxwell-type with a symmetric-hyperbolic formulation

Maxwell models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But the usual Maxwell models allow one to define well motions mostly for one-dimensional flows only. To define unequivocal multi-dimensional viscoelastic flows (as solutions to well-posed initial-value problems) we advocated in [ESAIM:M2AN 55 (2021), p. 807-831] an upper-convected Maxwell model for compressible flows with a symmetric-hyperbolic formulation. Here, that model is derived again, with new details.

Global attractor and l^p solutions to initial value problems of discrete nonlinear Schrodinger equations complex potential

In this article, we investigate the global well-posedness of initial value problems of the time-dependent discrete nonlinear Schrodinger equation with a complex potential and sufficiently general nonlinearity on a multidimensional lattice in weighted \( l^p\) spaces for \( 1< p <\infty\). Thanks to our improved estimates we are able to prove the existence of global attractor for \( l^p\) solutions to the initial value problem.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/12/abstr.html

The Fokas-Lenells equation on the line: Global well-posedness with solitons

In this paper, we prove the existence of global solutions in H3(R)∩H2,1(R) to the Fokas-Lenells (FL) equation on the line when the initial data includes solitons. A key tool in proving this result is a newly modified Darboux transformation, which adds or subtracts a soliton with given spectral and scattering parameters. In this way the inverse scattering transform technique is then applied to establish the global well-posedness of initial value problem with a finite number of solitons based on our previous results on the global well-posedness of the FL equation.

Existence and uniqueness of positive solution of a nonlinear differential equation with higher order Erdélyi-Kober operators

<abstract><p>In this paper, the initial value problem of a nonlinear differential equation with higher order Caputo type modification of the Erdélyi-Kober fractional derivatives was studied. Based on the transmutation method, the well-posedness of initial value problem of the higher order linear model was proved and an explicit solution was presented. Then some new Gronwall type inequalities involving Erdélyi-Kober fractional integral were established. By applying these results and some fixed point theorems, the existence and uniqueness of the positive solution of the nonlinear differential equation were proved. The method is applicable to the fractional differential equation with any order $ \gamma\in (n-1, n] $.</p></abstract>

Well-posed formulation of Einstein-Maxwell effective field theory

We consider the well-posedness of the initial value problem for Einstein-Maxwell theory modified by higher derivative effective field theory corrections. Field redefinitions can be used to bring the leading parity-symmetric 4-derivative corrections to a form which gives second order equations of motion. We show that a recently introduced "modified harmonic" gauge condition can be used to obtain a formulation of these theories which admits a well-posed initial value problem when the higher derivative corrections to the equations of motion are small.

Well-posedness, ill-posedness, and traveling waves for models of pulsatile flow in viscoelastic vessels

We study dispersive models of fluid flow in viscoelastic vessels, derived in the study of blood flow. The unknowns in the models are the velocity of the fluid in the axial direction and the displacement of the vessel wall from rest. We prove that one such model has a well-posed initial value problem, while we argue that a related model instead has an ill-posed initial value problem; in the second case, we still prove the existence of solutions in analytic function spaces. Finally we prove the existence of some periodic traveling waves.

Numerical relativity for Horndeski gravity

In this paper, we present an overview of recent developments in the numerical solution of Horndeski gravity theories, which are the class of all scalar–tensor theories of gravity that have second-order equations of motion. We review several methods that have been used to establish well-posed initial value problems for these theories, and discuss well-posed formulations of the constraint equations. We also discuss global aspects of exact, strongly coupled solutions to some of the Horndeski gravity theories: the formation of shocks, the loss of hyperbolicity, and the formation of naked curvature singularities. Finally, we discuss numerical solutions to binary black hole and neutron star systems for several Horndeski theories.

Fate of twin stars on the unstable branch: Implications for the formation of twin stars

Hybrid hadron-quark equations of state that give rise a third family of stable compact stars have been shown to be compatible with the LIGO-Virgo event GW170817. Stable configurations in the third family are called hybrid hadron-quark stars. The equilibrium stable hybrid hadron-quark star branch is separated by the stable neutron star branch with a branch of unstable hybrid hadron-quark stars. The end-state of these unstable configurations has not been studied, yet, and it could have implications for the formation and existence of twin stars -- hybrid stars with the same mass as neutron stars but different radii. We modify existing hybrid hadron-quark equations of state with a first-order phase transition in order to guarantee a well-posed initial value problem of the equations of general relativistic hydrodynamics, and study the dynamics of non-rotating or rotating unstable twin stars via 3-dimensional simulations in full general relativity. We find that unstable twin stars naturally migrate toward the hadronic branch. Before settling into the hadronic regime, these stars undergo (quasi)radial oscillations on a dynamical timescale while the core bounces between the two phases. Our study suggests that it may be difficult to form stable twin stars if the phase transition is sustained over a large jump in energy density, and hence it may be more likely that astrophysical hybrid hadron-quark stars have masses above the twin star regime. We also study the minimum-mass instability for hybrid stars, and find that these configurations do not explode, unlike the minimum-mass instability for neutron stars. Additionally, our results suggest that oscillations between the two Quantum Chromodynamic phases could provide gravitational wave signals associated with such phase transitions in core-collapse supernovae and white dwarf-neutron star mergers.

ECO-spotting: looking for extremely compact objects with bosonic fields

Black holes are thought to describe the geometry of massive, dark compact objects in the Universe. To further support and quantify this long-held belief requires knowledge of possible, if exotic alternatives. Here, we wish to understand how compact can self-gravitating solutions be. We discuss theories with a well-posed initial value problem, consisting in either a single self-interacting scalar, vector or both. We focus on spherically symmetric solutions, investigating the influence of self-interacting potentials into the compactness of the solutions, in particular those that allow for flat-spacetime solutions. We are able to connect such stars to hairy black hole solutions, which emerge as a zero-mass black hole. We show that such stars can have light rings, but their compactness is never parametrically close to that of black holes. The challenge of finding black hole mimickers to investigate full numerical-relativity binary setups remains open.

Well-posedness of initial value problem of Hirota-Satsuma system in low regularity Sobolev space

<abstract><p>In this paper, we study the initial value problem of Hirota-Satsuma system:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \notag \left\{ \begin{array}{ll} u_t-\alpha(u_{xxx}+6uu_x) = 2\beta vv_x, & \ x\in {\mathbb{R}}, \ t\ge 0, \\ v_t+v_{xxx}+3uv_x = 0, & x\in {\mathbb{R}}, \ t\ge 0, \\ u(0, x) = \phi(x), \; \; v(0, x) = \psi(x), & x\in {\mathbb{R}}, \end{array} \right. \end{equation} $\end{document} </tex-math></disp-formula></p> <p>where $ \alpha\in {\mathbb{R}} $, $ \beta\in {\mathbb{R}} $; $ u = u(x, t) $, $ v = v(x, t) $ are real functions. Aided by Fourier restrict norm method, we show that $ \forall s > -\frac 18 $ initial value problem (0.1) is locally well-posed in $ H^s({\mathbb{R}})\times H^{s+1}({\mathbb{R}}) $ which improved the results of <sup>[<xref ref-type="bibr" rid="b7">7</xref>]</sup>.</p></abstract>

Integrable Boundary Conditions for Quad Equations, Open Boundary Reductions, and Integrable Mappings

Abstract In the context of integrable difference equations on quad-graphs, we introduce the method of open boundary reductions, as an alternative to the well-known periodic reductions, for constructing discrete integrable mappings and their invariants. The mappings are obtained from well-posed initial value problems for quad and boundary equations restricted to strips on ${{\mathbb{Z}}}^2$-lattices. The invariants are constructed using Sklyanin’s double-row monodromy matrix. To establish its properties, we use the discrete zero curvature condition and boundary zero curvature condition, showing how the latter derives from the boundary consistency condition. We focus on the Adler–Bobenko–Suris classification and associated integrable boundary equations. Examples are given for the H1 and Q1($\delta =0$) equations, leading to novel maps of the plane.

Solving a well-posed fractional initial value problem by a complex approach

Nonlinear fractional differential equations have been intensely studied using fixed point theorems on various different function spaces. Here we combine fixed point theory with complex analysis, considering spaces of analytic functions and the behaviour of complex powers. It is necessary to study carefully the initial value properties of Riemann–Liouville fractional derivatives in order to set up an appropriate initial value problem, since some such problems considered in the literature are not well-posed due to their initial conditions. The problem that emerges turns out to be dimensionally consistent in an unexpected way, and therefore suitable for applications too.

Well-Posedness and Spectral Analysis of Integrodifferential Equations of Hereditary Mechanics

The well-posedness of initial value problems for abstract integrodifferential equations with unbounded operator coefficients in Hilbert spaces is studied, and spectral analysis of the operator functions that are the symbols of these equations is performed. The equations under consideration are an abstract form of linear partial integrodifferential equations arising in viscoelasticity theory, which have a number of other important applications. Results concerning the well-posedness of these integrodifferential equations in weighted Sobolev spaces of vector functions defined on the positive half-line with values in a Hilbert space are obtained. The localization and structure of the spectrum of the operator functions that are the symbols of these equations are established.

Well-posed formulation of Lovelock and Horndeski theories

We study the initial value problem for Lovelock and Horndeski theories of gravity. We show that the equations of motion of these theories can be written in a form that, at weak coupling, is strongly hyperbolic and therefore admits a well-posed initial value problem. This is achieved by introducing a new class of "modified harmonic" gauges for general relativity.

On Local Singularities in Ideal Potential Flows with Free Surface

Despite important advances in the mathematical analysis of the Euler equations for water waves, especially over the last two decades, it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems. For ideal free-surface flow with zero surface tension and gravity, the authors review existing works that describe “splash singularities”, singular hyperbolic solutions related to jet formation and “flip-through”, and a recent construction of a singular free surface by Zubarev and Karabut that however involves unbounded negative pressure. The authors illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation. Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data.

Spectral Analysis of Integrodifferential Equations in Hilbert Spaces

We investigate the well-posedness of initial-value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space and provide the spectral analysis of operator-functions describing symbols of such equations. These equations are an abstract form of linear partial integrodifferential equations arising in the viscoelasticity theory and other important applications. We establish the localization and the spectrum structure of operator-functions describing symbols of these equations.

Hyperbolicity of divergence cleaning and vector potential formulations of general relativistic magnetohydrodynamics

We examine hyperbolicity of general relativistic magnetohydrodynamics with divergence cleaning, a flux-balance law form of the model not covered by our earlier analysis. The calculations rely again on a dual-frame approach, which allows us to effectively exploit the structure present in the principal part. We find, in contrast to the standard flux-balance law form of the equations, that this formulation is strongly hyperbolic, and thus admits a well-posed initial value problem. Formulations involving the vector potential as an evolved quantity are then considered. Carefully reducing to first order, we find that such formulations can also be made strongly hyperbolic. Despite the unwieldy form of the characteristic variables we therefore conclude that of the free-evolution formulations of general relativistic magnetohydrodynamics presently used in numerical relativity, the divergence cleaning and vector potential formulations are preferred.

Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves

Truncated series models of gravity water waves are popular for use in simulation. Recent work has shown that these models need not inherit the well-posedness properties of the full equations of motion (the irrotational, incompressible Euler equations). We show that if one adds a sufficiently strong dispersive term to a quadratic truncated series model, the system then has a well-posed initial value problem. Such dispersion can be relevant in certain physical contexts, such as in the case of a bending force present at the free surface, as in a hydroelastic sheet.

Determination of thermal conductivity of inhomogeneous orthotropic materials from temperature measurements

ABSTRACTWe consider the two-dimensional inverse determination of the thermal conductivity of inhomogeneous orthotropic materials from internal temperature measurements. The inverse problem is general and is classified as a function estimation since no prior information about the functional form of the thermal conductivity is assumed in the inverse calculation. The least-squares functional minimizing naturally the gap between the measured and computed temperature leads to a set of direct, sensitivity and adjoint problems, which have forms of direct well-posed initial boundary value problems for the heat equation, and new formulas for its gradients are derived. The conjugate gradient method employs recursively the solution of these problems at each iteration. Stopping the iterations according to the discrepancy principle criterion yields a stable solution. The employment of the Sobolev -gradient is shown to result in much more robust and accurate numerical reconstructions than when the standard -gradient is used.

Well-Posedness and Spectral Analysis of Integrodifferential Equations Arising in Viscoelasticity Theory

We study the well-posedness of initial-value problems for abstract integrodifferential equations with unbounded operator coefficients in Hilbert spaces and provide a spectral analysis of operator functions that are symbols of the specified equations. These equations represent an abstract form of linear partial integrodifferential equations arising in viscoelasticity theory and other important applications. For the said integrodifferential equations, we obtain well-posedness results in weighted Sobolev spaces of vector functions defined on the positive semiaxis and valued in a Hilbert space. For the symbols of the said equations, we find the localization and the structure of the spectrum.