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A system of fractional differential equations (FDEs) with fractional derivatives of diverse orders is called an incommensurate system of FDEs. In this paper, the well-posedness of the initial value problem for incommensurate systems of FDEs is obtained on the space of continuous functions. Three different methods for this analysis are used and compared. The complexity of such analysis is reduced by new techniques. Strong existence results are obtained by weaker conditions. The uniqueness and the continuous dependency of the solution on initial values are investigated using the Gronwall inequality.

We consider a class of conditionally well-posed inverse problems characterized by a Hölder estimate of conditional stability on a convex compact in a Hilbert space. The input data and the operator of the forward problem are available with errors. We investigate the discretized residual functional constructed according to a general scheme of finite dimensional approximation. We prove that each its stationary point that is not too far from the finite dimensional approximation of the solution to the original inverse problem, generates an approximation from a small neighborhood of this solution. The diameter of the specified neighborhood is estimated in terms of characteristics of the approximation scheme. This partially removes iterating over local minimizers of the residual functional when implementing the discrete quasi-solution method for solving the inverse problem. The developed theory is illustrated by numerical examples.

Here, we study boundary value problems for the Laplace–Beltrami operator on a three-dimensional sphere with a circular cut, obtained by removing a smooth closed geodesic from S3 embedded in R4. The presence of the cut introduces singular perturbations of the domain, and we develop an analytical framework to characterize well-posed problems in this setting. Our approach combines Green’s functions, spectral analysis, and Sobolev space methods to establish solvability criteria and uniqueness results. In particular, we identify explicit conditions for the existence of solutions with data supported near the cut, and extend the formulation to include delta-type perturbations supported on the removed circle. These results generalize earlier work on punctured two-dimensional spheres and provide a foundation for the study of PDEs on manifolds with localized singularities.

In a number of previous works it was found that for binomial functional equations of the form a x u α x - λ u ( x ) = v ( x ) , x ∈ X , where α : X → X is an invertible map of the set X into itself, a situation typical for differential equations is possible: the equation is solvable for any right-hand side and there is no uniqueness of the solution. As in the case of differential equations, the question arises of formulating well-posed boundary-value problems, i.e., of specifying additional conditions under which the solution exists and is unique. In this paper, we discuss the question of what kind of additional conditions lead to well-posed boundary-value problems for the equations under consideration.

A class of conditionally well-posed problems characterized by a Hölder conditional stability estimate on a convex compact set in a Hilbert space is considered. The operator of the direct problem and the right-hand side of the equation are given with errors, and the derivatives of the exact and perturbed operators are not assumed to be close to each other. The convexity and single-extremality of the residual functional of the quasi-solution method are examined. For this functional, each of its stationary points on the set of conditional well-posedness that lies not too far from the sought solution of the original inverse problem is shown to belong to a small neighborhood of the solution. The diameter of this neighborhood is estimated in terms of the errors in the input data. It is shown that this neighborhood is an attractor of the iterations of the gradient projection method, and the convergence rate of the iterations to the attractor is estimated. The necessity of the used conditional stability estimate for the existence of iterative processes with the indicated properties is established.

We prove that the notion of Tykhonov well-posed problems is stable under the operation of inf-convolution. We deal with lower semicontinuous functions (not necessarily convex) defined on a metric magma. Several applications are given, in particular to the study of the map argmin.

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however, a major difficulty for this problem is the existence of non-integrable singularities of the function q y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann–Hilbert problem to an equivalent modified Riemann–Hilbert problem, we show that the solution can be expressed in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends explicitly on the width of the semistrip L , on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h ( λ ). The determination of the function h remains open.

The Benjamin–Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces Hs for s>−12. The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of Gérard’s explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy.

We establish the global well-posedness of strong solutions for the inhomogeneous magnetohydrodynamic equations with density-dependent viscosity in three-dimensional bounded domains, where the initial density is permitted to vanish. By skillfully employing various inequalities and scaling techniques, we demonstrate the existence and uniqueness of a global strong solution provided that ||σ( ρ 0 )|| L q is suitably small. Furthermore, we derive exponential decay rates for the solution. Importantly, no compatibility conditions need to be imposed on the initial data, even when vacuum states are present.

We study the Muskat problem for one fluid or two fluids, with or without viscosity jump, with or without rigid boundaries, and in arbitrary space dimension d of the interface. The Muskat problem is scaling invariant in the Sobolev space Hsc(Rd) where sc=1+d2. Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev spaces Hs(Rd), s>sc. Moreover, the rigid boundaries are only required to be Lipschitz and can have arbitrarily large variation. The Rayleigh–Taylor stability condition is assumed for the case of two fluids with viscosity jump but is proved to be automatically satisfied for the case of one fluid. The starting point of this work is a reformulation solely in terms of the Drichlet–Neumann operator. The key elements of proofs are new paralinearization and contraction results for the Drichlet–Neumann operator in rough domains.