Well-posedness Results Research Articles

On Singular Vortex Patches, I: Well-posedness Issues

The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globallymm-fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long asm≥3.m\geq 3.In this case, all of the angles involved solve aclosedODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness side, we show thatanyother type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angleπ2\frac {\pi }{2}for all time. Even in the case of vortex patches with corners of angleπ2\frac {\pi }{2}or with corners which are only locallymm-fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches onR2\mathbb {R}^2with interesting dynamical behavior such as cusping and spiral formation in infinite time.

The reverse Hölder inequality for matrix-valued stochastic exponentials and applications to quadratic BSDE systems

In this paper, we study the connections between three concepts — the reverse Hölder inequality for matrix-valued martingales, the well-posedness of linear BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE systems. In particular, we show that a linear BSDE with bmo coefficients is well-posed if and only if the stochastic exponential of a related matrix-valued martingale satisfies a reverse Hölder inequality. Furthermore, we give structural conditions under which these equivalent conditions are satisfied. Finally, we apply our results on linear equations to obtain global well-posedness results for two new classes of non-Markovian quadratic BSDE systems with special structure.

Nonlocal fully nonlinear parabolic differential equations arising in time-inconsistent problems

We prove the local well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter t on top of the temporal and spatial variables (s,y) and thus the problem could be considered as a flow of equations. The nonlocality comes from the dependence on the unknown function and its first- and second-order derivatives evaluated at not only the local point (t,s,y) but also at the diagonal line of the time domain (s,s,y). Such equations arise from time-inconsistent problems in game theory or behavioral economics, where the observations and preferences are (reference-)time-dependent. We first study the linearized version of the nonlocal PDEs with an innovative construction of appropriate norms and Banach spaces and contraction mappings over which. With fixed-point arguments, we obtain the solvability of nonlocal linear PDEs and establish a Schauder-type prior estimate for the solutions. Then, by the linearization method, we establish the well-posedness under the fully nonlinear case. Moreover, we reveal that the solution of a nonlocal fully nonlinear parabolic PDE is an adapted solution to a flow of second-order forward-backward stochastic differential equations.

Local well-posedness of a nonlinear Fokker–Planck model

Noise or fluctuations play an important role in the modeling and understanding of the behavior of various complex systems in nature. Fokker–Planck equations are powerful mathematical tools to study behavior of such systems subjected to fluctuations. In this paper we establish local well-posedness result of a new nonlinear Fokker–Planck equation. Such equations appear in the modeling of the grain boundary dynamics during microstructure evolution in the polycrystalline materials and obey special energy laws.

Rates of convergence for Laplacian semi-supervised learning with low labeling rates

We investigate graph-based Laplacian semi-supervised learning at low labeling rates (ratios of labeled to total number of data points) and establish a threshold for the learning to be well posed. Laplacian learning uses harmonic extension on a graph to propagate labels. It is known that when the number of labeled data points is finite while the number of unlabeled data points tends to infinity, the Laplacian learning becomes degenerate and the solutions become roughly constant with a spike at each labeled data point. In this work, we allow the number of labeled data points to grow to infinity as the total number of data points grows. We show that for a random geometric graph with length scale $$\varepsilon >0$$ , if the labeling rate $$\beta \ll \varepsilon ^2$$ , then the solution becomes degenerate and spikes form. On the other hand, if $$\beta \gg \varepsilon ^2$$ , then Laplacian learning is well-posed and consistent with a continuum Laplace equation. Furthermore, in the well-posed setting we prove quantitative error estimates of $$O(\varepsilon \beta ^{-1/2})$$ for the difference between the solutions of the discrete problem and continuum PDE, up to logarithmic factors. We also study p-Laplacian regularization and show the same degeneracy result when $$\beta \ll \varepsilon ^p$$ . The proofs of our well-posedness results use the random walk interpretation of Laplacian learning and PDE arguments, while the proofs of the ill-posedness results use $$\Gamma $$ -convergence tools from the calculus of variations. We also present numerical results on synthetic and real data to illustrate our results.

Stochastic integrodifferential models of fractional orders and Leffler nonsingular kernels: well-posedness theoretical results and Legendre Gauss spectral collocation approximations

Stochastic fractional integrodifferential models are widely employed to model several natural phenomena these days. This current work focuses on the well-posedness results and numerical solutions of a specific form of these models considering the Leffler nonsingular kernels operator wherein the stochastic term is driven by the standard Brownian motion. Accordingly, a combination of sufficient conditions, topological theorems, and Banach space theory are utilized to construct the well-posedness proof. For treating the numerical issue, a familiar spectral collocation technique relying upon shifted Legendre series expansion theory is proposed. The basic properties of Brownian motion and a linear spline interpolation method are used to simulate the standard Brownian motion at a fixed time value. In addition, the idea of the Gauss-Legendre numerical integration rule is implemented to approximate the finite integral. We also devote our attention to the concept of convergence of the proposed method and demonstrate its analysis. Ultimately, the obtained theoretical results and the presented method are examined with five numerous applications. The obtained results indicate the high accuracy and efficiency of applying this method in solving stochastic models of the above-mentioned form.

Mean-Field Type FBSDEs under Domination-Monotonicity Conditions and Application to LQ Problems

This paper is concerned with a class of mean-field type coupled forward-backward stochastic differential equations (MF-FBSDEs), in which the coupling appears in integral terms, terminal terms, and initial terms. Inspired by various mean-field type linear-quadratic (MF-LQ) optimal control problems, we propose a type of randomized domination-monotonicity conditions, under which and the usual Lipschitz condition we obtain a well-posedness result on MF-FBSDEs in the sense of square integrability including the unique solvability, an estimate of the solution, and the related continuous dependence property of the solution on the coefficients. The result of MF-FBSDEs in turn extends MF-LQ problems in the literature to a general situation where the initial states or the terminal states are also controlled at the same time, and gives explicit expressions of the related unique optimal controls.

Well-Posedness for the Incompressible Hall-MHD System with Initial Magnetic Field Belonging to $$H^{\frac{3}{2}}({\mathbb {R}}^3)$$

In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system for initial data $$(u_0,B_0)\in H^{\frac{1}{2}+\sigma }({\mathbb {R}}^3)\times H^{\frac{3}{2}}({\mathbb {R}}^3)$$ with $$\sigma \in (0,2)$$ . In particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce $$\sigma $$ to 0 with the aid of the new formulation of the Hall-MHD system observed by Danchin and Tan (Commun Partial Differ Equ 46(1):31–65, 2021). Compared with the previous works, our local well-posedness results improve the regularity condition on the initial data. Moreover, we establish the global well-posedness for small initial data in $$H^{\frac{1}{2}+\sigma }({\mathbb {R}}^3)\times H^{\frac{3}{2}}({\mathbb {R}}^3)$$ with $$\sigma \in (0,2)$$ , and get the optimal time-decay rates of solutions.

A new reformulation of the Muskat problem with surface tension

Two formulas that connect the derivatives of the double layer potential and of a related singular integral operator evaluated at some density ϑ to the L2-adjoints of these operators evaluated at the density ϑ′ are used to recast the Muskat problem with surface tension and general viscosities as a system of equations with nonlinearities expressed in terms of the L2-adjoints of these operators. An advantage of this formulation is that the nonlinearities appear now as a derivative. This aspect and abstract quasilinear parabolic theory are then exploited to establish a local well-posedness result in all subcritical Sobolev spaces Wps(R) with p∈(1,∞) and s∈(1+1/p,2).

Wellposedness of NLS in Modulation Spaces

We prove new local and global well-posedness results for the cubic one-dimensional nonlinear Schrödinger equation in modulation spaces. Local results are obtained via multilinear interpolation. Global results are proven using conserved quantities based on the complete integrability of the equation, persistence of regularity, and by separating off the time evolution of finitely many Picard iterates.

Nonlinear wave equations with slowly decaying initial data

New local smoothing estimates in Besov spaces adapted to the half-wave group are proved via ℓ2-decoupling. We apply these estimates to obtain new well-posedness results for the cubic nonlinear wave equation in two dimensions. The results are compared to new well-posedness results in Lp-based Sobolev spaces.

Analysis of a linearized poromechanics model for incompressible and nearly incompressible materials

In this work, we thoroughly analyze the linearized version of a poromechanics model developed to simulate soft tissues perfusion. This is a fully unsteady model in which the fluid and solid equations are strongly coupled through the interstitial pressure. As such, it generalizes Darcy, Brinkman and Biot equations of poroelasticity. The mathematical and numerical analysis of this model was initially performed for a compressible porous material. Here, we focus on the nearly incompressible case with a semigroup approach, which also allows us to prove the existence of weak solutions. We show the existence and uniqueness of strong and weak solutions in the incompressible limit case, for which a divergence constraint on the mixture velocity appears. Due to the special form of the coupling, the underlying problem is not coercive. Nevertheless, by using the notion of $ T $-coercivity, we obtain stability estimates and well-posedness results. Our study also provides guidelines to propose stable and robust approximations of the problem with mixed finite elements. In particular, we recover an inf-sup condition that is independent of the porosity. Finally, we numerically investigate the elliptic regularity of the associated steady-state problem and illustrate the sensitivity of the solution with respect to the different model parameters.

Radon measure-valued solutions of unsteady filtration equations

<p style='text-indent:20px;'>We study Radon measure-valued solutions of a Cauchy-Dirichlet problem in one space dimension that describes nonlinear convection-diffusion phenomena in heterogeneous media. Well-posedness results are proven, which extend those previously known for the space homogeneous case.</p>

On the singular limit of a boundary delayed Kuramoto-Sivashinsky-Korteweg-de Vries equation: Well-posedness and stability results

On the singular limit of a boundary delayed Kuramoto-Sivashinsky-Korteweg-de Vries equation: Well-posedness and stability results

Cone-bounded feedback laws for linear second order systems

The strong stabilizability of linear second order equation is investigated in this paper. It is supposed that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable. Then this control is subject to a cone-bounded nonlinearity. Well-posedness and stability results of the closed-loop system under such (nonlinear) control are stated. The first result is proven by using nonlinear semigroups techniques and the Schauder fixed-point theorem and the second one is based on the result of Marx et al. [12]. Our results are then applied to the particular damped and undamped wave equation. Simulation results are presented to validate the theoretical results. Note that some of the results of this paper apply for a large class systems.

Well-Posedness and Infinite Propagation Speed for the Fifth-Order Camassa-Holm Equation with Weakly Dissipative Term

In this paper, we study the fifth-order Camassa-Holm equation with weakly dissipative term. We first give the local well-posedness result and the blow up criterion. Then, we establish sufficient conditions to guarantee that the solution exists globally in time. Finally, the infinite propagation speed of this equation is also investigated.

Existence and well-posed results for nonclassical diffusion systems with nonlocal diffusion

Our goal in this paper is to present well-posed results for nonclassical diffusion systems which have applications in population dynamics. First, we establish the existence and uniqueness of a mild solution to the initial value problem. The asymptotic behavior of the mild solution is also considered when the parameter tends to zero. Second, we obtain a local well-posedness result for nonclassical diffusion systems with a nonlocal time condition. The main idea to obtain the above theoretical results is to use Banach?s theorem and some techniques in Fourier series analysis. Some numerical tests are also presented to illustrate the theory.

Pullback dynamics and robustness for the 3D Navier-Stokes-Voigt equations with memory

<abstract><p>The tempered pullback dynamics and robustness of the 3D Navier-Stokes-Voigt equations with memory and perturbed external force are considered in this paper. Based on the global well-posedness results and energy estimates involving memory, a suitable tempered universe is constructed, the robustness is finally established via the upper semi-continuity of tempered pullback attractors when the perturbation parameter epsilon tends to zero.</p></abstract>

Remarks on the global well-posedness of the axisymmetric Boussinesq system with rough initial data

This work establishes a global well-posedness result for the three dimensional axisymmetric Boussinesq system with critical rough initial data. Specifically, we aim at extending the results in our previous paper [22] to the case where the initial vorticity and density are finite Radon measures. The roadmap towards proving our claim is based upon the strategy in [22, 17]. Namely, performing a fixed point argument to build up a unique local solution, then, showing that the constructed solution is in fact global in time. Nevertheless, crucial arguments from [22], required for the construction of the local solution, need to be carefully analyzed when the datum, and more precisely when the initial density, is not a Lebesgue-integrable function. This is where several techniques of measure theory come into play. Accordingly, we first develop numerous notions on axisymmetric measures, in a general context. These techniques are employed thereafter to understand the coupling between some important quantities of the system and, eventually, to construct the desired solution whenever only the atomic parts of the initial measures are small enough.

The p-curl system in extension domains

In this paper we prove well-posedness results for a parabolic $ p $-$ {\rm{curl}}\; $ system on a three-dimensional bounded extension domain. In view of the numerical approximation, we investigate the asymptotic behavior of the solutions to suitable approximating problems. Crucial tools are a Stokes formula and a Gaffney inequality for extensions domains.