Smooth Initial Data Research Articles

Energy bounds for a fourth-order equation in low dimensions related to wave maps

For compact, isometrically embedded Riemannian manifolds N ↪ R L N \hookrightarrow \mathbb {R}^L , we introduce a fourth-order version of the wave maps equation. By energy estimates, we prove an a priori estimate for smooth local solutions in the energy subcritical dimension n = 1 n = 1 , 2 2 . The estimate excludes blow-up of a Sobolev norm in finite existence times. In particular, combining this with recent work of local well-posedness of the Cauchy problem, it follows that for smooth initial data with compact support, there exists a (smooth) unique global solution in dimension n = 1 n=1 , 2 2 . We also give a proof of the uniqueness of solutions that are bounded in these Sobolev norms.

On the development of shocks in fluids

In my 2007 work, I studied the formation of shocks in the context of Eulerian equations of the mechanics of compressible fluids. That work studied the maximal classical development of smooth initial data. The present review article is a presentation of my 2019 work, which addresses the problem of the physical continuation of the solution past the point of shock formation. The problem requires the construction of a hypersurface, the shock hypersurface, originating from a given singular spacelike surface, which is acoustically spacelike as viewed from its past, and the construction of a new solution in the future of the union of a given regular null hypersurface with the shock hypersurface, a solution which extends continuously the prior maximal classical solution across the given regular null hypersurface, but which displays discontinuities in physical variables across the shock hypersurface in accordance with the mass, momentum, and energy conservation laws. Moreover, the shock hypersurface is to be acoustically timelike as viewed from its future. Mathematically, this is a free boundary problem, with nonlinear conditions at the free boundary, for a first order quasilinear hyperbolic system of PDE, with characteristic initial data, which are singular at the past boundary of the initial null hypersurface, that boundary being the singular surface of origin. My 2019 work solved a restricted form of the problem through the introduction of new geometric and analytic methods. This Review is focused on the presentation of these methods.

On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles

In this paper and its sequel, we construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible three-dimensional Navier-Stokes and Euler equations implode (with infinite density) at a later time at a point, and we completely describe the associated formation of singularity. This paper is concerned with existence of smooth self-similar profiles for the barotropic Euler equations in dimension $d\ge 2$ with decaying density at spatial infinity. The phase portrait of the nonlinear ODE governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. It allows us to construct global profiles of the self-similar problem, which however turn out to be generically non-smooth across the associated acoustic cone. In a suitable range of barotropic laws and for a sequence of quantized speeds accumulating to a critical value, we prove the existence of non-generic $\mathcal{C}^\infty$ self-similar solutions with suitable decay at infinity. The $\mathcal{C}^\infty$ regularity is used in a fundamental way in our companion paper (part II) in the analysis of the associated linearized operator and leads, in turn, to the construction of finite energy blow up solutions of the compressible Euler and Navier-Stokes equations in dimensions $d=2,3$.

On the implosion of a compressible fluid II: Singularity formation

In this paper, which continues our investigation of strong singularity formation in compressible fluids, we consider the compressible three-dimensional Navier-Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. An essential step in the proof is the existence of $\mathcal{C}^\infty$ smooth self-similar solutions to the compressible Euler equations for quantized values of the speed constructed in our companion paper (part I). All blow up dynamics obtained for the Navier-Stokes problem are of type II (non self-similar).

Finite time blow-up for some parabolic systems arising in turbulence theory

We study a class of non-linear parabolic systems relevant in turbulence theory. Those systems can be viewed as simplified versions of the Prandtl one-equation and Kolmogorov two-equation models of turbulence. We restrict our attention to the case of one space dimension. We consider initial data for which the diffusion coefficients may vanish. We prove that, under this condition, those systems are locally well-posed in the class of Sobolev spaces of high enough regularity, but also that there exist smooth initial data for which the corresponding solutions blow up in finite time. We are able to put in evidence two different types of blow-up mechanism. In addition, the results are extended to the case of transport-diffusion systems, namely to the case when convection is taken into account.

Nonsmooth data error estimates of the L1 scheme for subdiffusion equations with positive-type memory term

Abstract This paper considers fully discrete finite element approximations to subdiffusion equations with memory in a bounded convex polygonal domain. We first derive some regularity results for the solution with respect to both smooth and nonsmooth initial data in various Sobolev norms. These regularity estimates cover the cases when $u_0\in L^2(\varOmega )$ and the source function is Hölder continuous in time. The spatially discrete scheme is developed using piecewise linear and continuous finite elements, and optimal-order error bounds for both homogeneous and nonhomogeneous problems are established. The temporal discretization based on the L1 scheme is considered and analyzed. We prove optimal error estimates in time for both homogeneous and nonhomogeneous problems. Finally, numerical results are provided to support our theoretical analysis.

Generalized Persistence of Entropy Weak Solutions for System of Hyperbolic Conservation Laws

Let u(t, x) be the solution to the Cauchy problem of a scalar conservation law in one space dimension. It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution can best lie in the space of bounded total variations. It is impossible that the solutions belong to, for example, H1 because by Sobolev embedding theorem H1 functions are Hölder continuous. However, the author notes that from any point (t, x), he can draw a generalized characteristic downward which meets the initial axis at y = α(t, x). If he regards u as a function of (t, y), it indeed belongs to H1 as a function of y if the initial data belongs to H1. He may call this generalized persistence (of high regularity) of the entropy weak solutions. The main purpose of this paper is to prove some kinds of generalized persistence (of high regularity) for the scalar and 2 × 2 Temple system of hyperbolic conservation laws in one space dimension.

Global classical solutions to the compressible micropolar viscous fluids with large oscillations and vacuum

In this paper, we consider the three dimensional Cauchy problem of the compressible micropolar viscous flows. We prove the existence of unique global classical solution for smooth initial data with small initial energy but possibly large oscillations and the initial density may allowed to contain the interior and far field vacuum states. Furthermore, the large time behavior of the solution is obtained as well.

Global generalized solutions for a two-species chemotaxis system with tensor-valued sensitivity and logistic source

This paper is devoted to investigating a two-species chemotaxis system with tensor-valued sensitivity and logistic source. It is mainly concerned with the global existence of generalized solutions. More precisely, when the logistic-type is weaker than the usual quadratic case, we prove, for any sufficiently smooth initial data, the associated no-flux/no-flux/no-flux problem possesses at least one globally defined solution in an appropriate generalized sense. In addition, we also underline that the similar results have been established for the corresponding system with constant sensitivity in a well-known result if the degradation is quadratic. Our results show that logistic source can rule out blow-up for such kind of models. On the other hand, due to the introduction of tensor value sensitivity, we neither need to impose any smallness assumption on the initial data here, nor require any restriction on the spatial dimension which generalizes and improves the partial of the result in the literature.

On the 3D Incompressible Boussinesq Equations in a Class of Variant Spherical Coordinates

This paper investigates the global stabilizing effects of the geometry of the domain at which the flow locates and of the geometric structure of the solution to the incompressible flows by studying the three-dimensional (3D) incompressible, viscosity, and diffusivity Boussinesq system in spherical coordinates. We establish the global existence and uniqueness of the smooth solution to the Cauchy problem for a full 3D incompressible Boussinesq system in a class of variant spherical coordinates for a class of smooth large initial data. We also construct one class of nonempty bounded domains in the three-dimensional space ℝ 3 , in which the initial boundary value problem for the full 3D Boussinesq system in a class of variant spherical coordinates with a class of large smooth initial data with swirl has a unique global strong or smooth solution with exponential decay rate in time.

Formation of Point Shocks for 3D Compressible Euler

We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove that for an open set of Sobolev‐class initial data that are a small L∞ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blowup time and location can be explicitly computed, and solutions at the blowup time are smooth except for a single point, where they are of cusp‐type with Hölder C1/3 regularity. Our proof is based on the use of modulated self‐similar variables that are used to enforce a number of constraints on the blowup profile, necessary to establish global existence and asymptotic stability in self‐similar variables. © 2022 Wiley Periodicals LLC.

Shock Formation and Vorticity Creation for 3d Euler

AbstractWe analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3, 4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the nonisentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direction. This is achieved by establishing the asymptotic stability of a generic shock profile in modulated self‐similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high‐order energy estimates in Sobolev spaces. © 2022 Wiley Periodicals LLC.

Global Regularity for a Nonlocal PDE Describing Evolution of Polynomial Roots Under Differentiation

In this paper, we analyze a nonlocal nonlinear partial differential equation formally derived by Steinerberger [Proc. Amer. Math. Soc., 147 (2019), pp. 4733--4744] to model dynamics of roots of polynomials under differentiation. This partial differential equation is critical and bears striking resemblance to hydrodynamic models used to describe collective behavior of agents (such as birds, fish, or robots) in mathematical biology. We consider a periodic setting and show global regularity and exponential in time convergence to uniform density for solutions corresponding to strictly positive smooth initial data.

Large-data global existence in a higher-dimensional doubly degenerate nutrient system

This paper deals with a doubly degenerate nutrient taxis system given by(⋆){ut=∇⋅(uv∇u)−χ∇⋅(uαv∇v)+ℓuv,x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0, in a smoothly bounded convex domain Ω⊂Rn, n=2,3, with zero-flux boundary conditions, where α>0,χ>0 and ℓ≥0.It is known that when α=2, for which, in fact, the system was proposed by Leyva et al. (2013) to model the aggregation patterns of colonies of Bacillus subtilis identified on the surface of thin agar plates, a globally defined weak solution to (⋆) can be obtained, either in the case n=1, or for its corresponding spatially two-dimensional version, the latter case additionally requiring certain restrictions on the size of initial data. The present paper complements the results, asserting that under the assumptionα∈{(1,32)if n=2,and(76,139)if n=3, for all sufficiently smooth initial data, (⋆) possesses such a global weak solution (u,v).

Local strict singular characteristics: Cauchy problem with smooth initial data

Main purpose of this paper is to study the local propagation of singularities of viscosity solution to contact type evolutionary Hamilton-Jacobi equationDtu(t,x)+H(t,x,Dxu(t,x),u(t,x))=0. An important issue of this topic is the existence, uniqueness and regularity of the strict singular characteristic. We apply the recent existence and regularity results on the Herglotz' type variational problem to the aforementioned Hamilton-Jacobi equation with smooth initial data. We obtain some new results on the local structure of the cut set of the viscosity solution near non-conjugate singular points. Especially, we obtain an existence result of smooth strict singular characteristic from and to non-conjugate singular initial point based on the structure of the superdifferential of the solution, which is even new in the classical time-dependent case. We also get a global propagation result for the C1 singular support in the contact case.

Norm-inflation results for purely BBM-type Boussinesq systems

This article is concerned with the norm-inflation phenomena associated with a periodic initial-value abcd-Benjamin-Bona-Mahony type Boussinesq system. We show that the initial-value problem is ill-posed in the periodic Sobolev spaces Hp−s(0,2π)×Hp−s(0,2π) for all s>0. Our proof is constructive, in the sense that we provide smooth initial data that generates solutions arbitrarily large in Hp−s(0,2π)×Hp−s(0,2π)-norm for arbitrarily short time. This result is sharp since in [13] the well-posedness is proved to holding for all positive periodic Sobolev indexes of the form Hps(0,2π)×Hps(0,2π), including s=0.

Rannacher time-marching with orthogonal spline collocation method for retrieving the discontinuous behavior of hedging parameters

In this paper, we extensively study the orthogonal spline collocation method known as the spline collocation at Gauss points with Rannacher’s time-marching scheme for free boundary value option pricing problems. Such financial problems commonly feature non-smooth payoff functions that cause inaccuracies in approximating the solution and its derivatives. As a result, unlike the problems with the smooth initial data, the quadratic convergence is not realized by the Crank-Nicolson time-stepping scheme for these problems. Furthermore, the non-smoothness in the initial condition leads to severe degradation in the convergence rates and spurious oscillations near the discontinuity. The rationale is that classical schemes strongly rely on the smoothness of the initial data. A rigorous time-marching scheme referred to as Rannacher time-stepping scheme is introduced for the American option’s price diagnosed by a linear complementarity problem to smoothen the data. Moreover, with careful analysis, second and fourth orders of convergence are established for the present scheme in temporal and spatial directions, respectively. The numerical results for three test problems are presented in tables and graphs to validate the theory. These results show that the present scheme achieves higher accuracy and sufficiently restores the expected behavior.

A potential two-scale traveling wave singularity for 3D incompressible Euler equations

In this paper, we investigate a potential two-scale traveling wave singularity of the 3D incompressible axisymmetric Euler equations with smooth initial data of finite energy. The two-scale feature is characterized by the property that the center of the traveling wave approaches to the origin at a slower rate than the rate of the collapse of the singularity. The driving mechanism for this potential singularity is due to two antisymmetric vortex dipoles that generate a strong shearing layer in both the radial and axial velocity fields. Without any viscous regularization, the 3D Euler equations develop an additional small scale characterizing the thickness of the sharp front. In order to stabilize the rapidly decreasing thickness of the sharp front, we apply a vanishing first order numerical viscosity to the Euler equations. We present numerical evidence that the 3D Euler equations with this first order numerical viscosity develop a locally self-similar blowup at the origin.

РЕГУЛЯРНОСТЬ И АППРОКСИМАЦИЯ РЕШЕНИЯ ОДНОСТОРОННЕЙ ЗАДАЧИ ДЛЯ ПСЕВДОПАРАБОЛИЧЕСКОГО ОПЕРАТОРА БАРЕНБЛАТТА – ЖЕЛТОВА – КОЧИНОЙ

We consider a one-sided problem for the Barenblatt–Zheltov–Kochina pseudoparabolic operator in the one-dimensional case, supplemented with smooth initial data and homogeneous boundary conditions. This problem is formulated in the form of a variational inequality. From the physical point of view, it models a non-stationary process of filtration of a viscous fluid in a cracky-porous gallery with a restriction on the modulus of the velocity of filtration through the cracks. The existence theorem for a weak solution of this problem is known in the literature in both one-dimensional and multidimensional cases and follows from the results obtained by M. Ptashnyk (Nonlinear Anal., 2007, vol. 66, pp. 2653–2675) using the penalty method. In M. Ptashnyk’s research, the penalty operator was chosen in a standard form, following the presentation in the monograph by J.-L. Lions “Quelques m´ethodes de r´esolution des probl´emes aux limites non lin´eaires,” Paris: Dunod Gauthier-Villars, 1969 (Theorem 5.1 in Chapter 3). In this article, we consider an approximate initial-boundary value problem for the pseudoparabolic equation incorporating Kaplan’s penalty operator and study the family of its solutions. Due to the specific structure of Kaplan’s operator, we obtain higher regularity of the weak solution of the original problem as compared to the previously known regularity properties, and also we find a strengthened property of approximating this solution by a sequence of solutions to the problem with Kaplan’s operator. In addition, we establish that the one-sided condition imposed in the original problem is satisfied by the approximate solution on a set of the spatial variable which monotone grows with decrease of the small approximation parameter.

Formation and Construction of a Multidimensional Shock Wave for the First-Order Hyperbolic Conservation Law with Smooth Initial Data

In this paper, the problem on formation and construction of a multidimensional shock wave is studied for the first order conservation law $\partial_t u+\partial_x F(u)+\partial_y G(u)=0$ with smooth initial data $u_0(x,y)$. It is well-known that the smooth solution $u$ will blow up on the time $T^*=-\frac{1}{\min{H(\xi,\eta)}}$ when $\min{H(\xi,\eta})<0$ holds for $H(\xi,\eta)=\partial_{\xi}(F'(u_0(\xi,\eta)))+\partial_{\eta}(G'(u_0(\xi,\eta)))$, more precisely, only the first order derivatives $\nabla_{t,x,y}u$ blow up on $t=T^*$ meanwhile $u$ itself is still continuous until $t=T^*$. Under the generic nondegenerate condition of $H(\xi,\eta)$, we construct a local weak entropy solution $u$ for $t\ge T^*$ which is not uniformly Lipschitz continuous on two sides of a shock surface $\Sigma$. The strength of the constructed shock is zero on the initial blowup curve $\Gamma$ and then gradually increases for $t>T^*$. Additionally, in the neighbourhood of $\Gamma$, some detailed and precise descriptions on the singularities of solution $u$ are given.