Criterion for the Solvability of the Weighted Cauchy Problem for an Abstract Euler–Poisson–Darboux Equation

In a Banach space E, we consider the abstract Euler–Poisson–Darboux equation u″(t) + kt^?1u′(t) = Au(t) on the half-line. (Here k ∈ ? is a parameter, and A is a closed linear operator with dense domain on E.) We obtain a necessary and sufficient condition for the solvability of the Cauchy problem u(0) = 0, lim _t→0+t ^k u′(t) = u₁, k < 0, for this equation. The condition is stated in terms of an estimate for the norms of the fractional power of the resolvent of A and its derivatives. We introduce the operator Bessel function with negative index and study its properties.     

Cauchy problems with modified conditions for the Euler–Poisson–Darboux equations in the hyperbolic space

In this note, we give the solutions of the Cauchy problems for the Euler–Poisson–Darboux equations (EPD) with modified conditions in the hyperbolic space with application to the wave equation.     

Boundary value problem for a generalized Cauchy–Riemann equation with singular coefficients

For a generalized Cauchy–Riemann system whose coefficients admit higher-order singularities on a segment, we obtain an integral representation of the general solution and study a boundary value problem combining the properties of the linear conjugation problem and the Riemann–Hilbert problem in function theory.     

The Cauchy problem for the dissipation-modified Kadomtsev–Petviashvili equation

Considered herein is the dissipation-modified Kadomtsev–Petviashvili equation in two space-dimensional case. It is established that the Cauchy problem associated to this equation is locally well-posed in anisotropic Sobolev spaces. It is also shown in some sense that this result is sharp. In addition, the global well-posedness for this equation under suitable conditions is proved.     

On the Cauchy problem for the two-component Euler–Poincaré equations

In the paper, we first use the energy method to establish the local well-posedness as well as blow-up criteria for the Cauchy problem on the two-component Euler–Poincaré equations in multi-dimensional space. In the case of dimensions 2 and 3, we show that for a large class of smooth initial data with some concentration property, the corresponding solutions blow up in finite time by using Constantin–Escher Lemma and Littlewood–Paley decomposition theory. Then for the one-component case, a more precise blow-up estimate and a global existence result are also established by using similar methods. Next, we investigate the zero density limit and the zero dispersion limit. At the end, we also briefly demonstrate a Liouville type theorem for the stationary weak solution.     

Cauchy–Gelfand problem and the inverse problem for a first-order quasilinear equation

Gelfand’s problem on the large time asymptotics of the solution of the Cauchy problem for a first-order quasilinear equation with initial conditions of the Riemann type is considered. Exact asymptotics in the Cauchy–Gelfand problem are obtained and the initial data parameters responsible for the localization of shock waves are described on the basis of the vanishing viscosity method with uniform estimates without the a priori monotonicity assumption for the initial data.     

A note on the exact discretization for a Cauchy–Euler equation: application to the Black–Scholes equation

We construct the exact finite difference representation for a second-order, linear, Cauchy–Euler ordinary differential equation. This result is then used to construct new non-standard finite difference schemes for the Black–Scholes partial differential equation.     

The Cauchy problem for the generalized Zakharov–Kuznetsov equation on modulation spaces

We consider the Cauchy problem for the generalized Zakharov–Kuznetsov equation ${?}_{t}u+{?}_{x_1}\mathrm{\Delta }u={?}_{x_1}(u^{m+1})$ on three and higher dimensions. We mainly study the local well-posedness and the small data global well-posedness in the modulation space $M_{2,1}^0({\mathbb{R}}^n)$ for $m\ge 4$ and $n\ge 3$. We also investigate the quartic case, i.e., $m=3$.     

A control type method for solving the Cauchy–Stokes problem

In this work, we propose an approximate optimal control formulation of the Cauchy problem for the Stokes system. Here the problem is converted into an optimization one. In order to handle the instability of the solution of this ill-posed problem, a regularization technique is developed. We add a term in the least square function which happens to vanish while the algorithm converges. The efficiency of the proposed method is illustrated by numerical experiments.     

On the Cauchy problem for the generalized Benjamin–Ono equation with small initial data

We prove global well-posedness results for small initial data in $\text{H}{}^{\mathrm{s}}\text{(}\text{R}\text{),}\text{s>s}{}_{\mathrm{k}}$, and in , s_k=1/2?1/k, for the generalized Benjamin–Ono equation $\text{?}{}_{\mathrm{t}}\text{u+}\text{H}\text{?}{}^2{}_{\mathrm{x}}\text{u+?}{}_{\mathrm{x}}\text{(u}{}^{\mathrm{k+1}}\text{)=0,}\text{k?4}$. We also consider the cases k=2,3. To cite this article: L. Molinet, F. Ribaud, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Instantaneous blow-up of classical solutions to the Cauchy problem for the Khokhlov–Zabolotskaya equation

The Cauchy problem for a second-order nonlinear equation with mixed derivatives is considered. It is proved that its classical local-in-time solution does not exist. The blow-up of the solution is proved by applying S.I. Pohozaev and E.L. Mitidieri’s nonlinear capacity method.     

On the Euler–Lagrange equation for a variational problem: the general case II

In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L^￥_loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem

inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      15. An explicit form of the solution of the Cauchy problem for the Barenblatt–Zheltov–Kochina equation      Kh. G. Umarov 《Journal of Applied Mathematics and Mechanics》2014,78(2):144-154 A solution of the Cauchy problem in an isotropic medium and in an anisotropic medium with clearly expressed vertical or horizontal permeability is constructed for the Barenblatt–Zheltov–Kochina model representation of the seepage of a fluid in fissured porous rock by reducing the seepage problems under consideration to solving an abstract Cauchy problem in a Banach space.      16. From the Vlasov–Poisson equation with strong local alignment to the pressureless Euler–Poisson system      《Applied Mathematics Letters》2018 This article provides a rigorous justification on a hydrodynamic limit from the Vlasov–Poisson system with strong local alignment to the pressureless Euler–Poisson system for repulsive dynamics.      17. Cauchy problem for the Fokker–Planck–Kolmogorov equation of a multidimensional normal Markovian process      S. D. Ivasyshen  H. S. Pasichnyk 《Journal of Mathematical Sciences》2011,176(4):505-514 We present the results of studying the fundamental solution and correct solvability of the Cauchy problem as well as the integral representation of solutions for the Fokker–Planck–Kolmogorov equation of a class of normal Markovian processes.      18. On the Cauchy problem of generalized Fokas–Olver–Resenau–Qiao equation      Meiling Yang  Yongsheng Li 《Applicable analysis》2018,97(13):2246-2268       19. Asymptotic limits of solutions to the initial boundary value problem for the relativistic Euler–Poisson equations      La-Su Mai  Hai-Liang Li  Kaijun Zhang 《Journal of Differential Equations》2014 We study the asymptotic limit problem on the relativistic Euler–Poisson equations. Under the assumptions of both the initial data being the small perturbation of the given steady state solution and the boundary strength being suitably small, we have the following results: (i) the global smooth solution of the relativistic Euler–Poisson equation converges to the solution of the drift-diffusion equations provided the light speed c and the relaxation time τ   satisfying c=τ−1/2$c={\tau }^{-1/2}$ when the relaxation time τ   tends to zero; (ii) the global smooth solution of the relativistic Euler–Poisson equations converges to the subsonic global smooth solution of the unipolar hydrodynamic model for semiconductors when the light speed c→∞$c\to \infty$. In addition, the related convergence rate results are also obtained.      20. The Stefan problem for the Fisher–KPP equation      Yihong Du  Zongming Guo 《Journal of Differential Equations》2012,253(3):996-1035 We study the Fisher–KPP equation with a free boundary governed by a one-phase Stefan condition. Such a problem arises in the modeling of the propagation of a new or invasive species, with the free boundary representing the propagation front. In one space dimension this problem was investigated in Du and Lin (2010) [11], and the radially symmetric case in higher space dimensions was studied in Du and Guo (2011) [10]. In both cases a spreading-vanishing dichotomy was established, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or fails to establish and dies out in the long run; moreover, in the case of spreading, the asymptotic spreading speed was determined. In this paper, we consider the non-radially symmetric case. In such a situation, similar to the classical Stefan problem, smooth solutions need not exist even if the initial data are smooth. We thus introduce and study the “weak solution” for a class of free boundary problems that include the Fisher–KPP as a special case. We establish the existence and uniqueness of the weak solution, and through suitable comparison arguments, we extend some of the results obtained earlier in Du and Lin (2010) [11] and Du and Guo (2011) [10] to this general case. We also show that the classical Aronson–Weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here.

An explicit form of the solution of the Cauchy problem for the Barenblatt–Zheltov–Kochina equation

A solution of the Cauchy problem in an isotropic medium and in an anisotropic medium with clearly expressed vertical or horizontal permeability is constructed for the Barenblatt–Zheltov–Kochina model representation of the seepage of a fluid in fissured porous rock by reducing the seepage problems under consideration to solving an abstract Cauchy problem in a Banach space.     

From the Vlasov–Poisson equation with strong local alignment to the pressureless Euler–Poisson system

This article provides a rigorous justification on a hydrodynamic limit from the Vlasov–Poisson system with strong local alignment to the pressureless Euler–Poisson system for repulsive dynamics.     

Cauchy problem for the Fokker–Planck–Kolmogorov equation of a multidimensional normal Markovian process

We present the results of studying the fundamental solution and correct solvability of the Cauchy problem as well as the integral representation of solutions for the Fokker–Planck–Kolmogorov equation of a class of normal Markovian processes.     

Asymptotic limits of solutions to the initial boundary value problem for the relativistic Euler–Poisson equations

We study the asymptotic limit problem on the relativistic Euler–Poisson equations. Under the assumptions of both the initial data being the small perturbation of the given steady state solution and the boundary strength being suitably small, we have the following results: (i) the global smooth solution of the relativistic Euler–Poisson equation converges to the solution of the drift-diffusion equations provided the light speed c and the relaxation time τ   satisfying c=τ^−1/2$c={\tau }^{-1/2}$ when the relaxation time τ   tends to zero; (ii) the global smooth solution of the relativistic Euler–Poisson equations converges to the subsonic global smooth solution of the unipolar hydrodynamic model for semiconductors when the light speed c→∞$c\to \infty$. In addition, the related convergence rate results are also obtained.     

The Stefan problem for the Fisher–KPP equation

We study the Fisher–KPP equation with a free boundary governed by a one-phase Stefan condition. Such a problem arises in the modeling of the propagation of a new or invasive species, with the free boundary representing the propagation front. In one space dimension this problem was investigated in Du and Lin (2010) [11], and the radially symmetric case in higher space dimensions was studied in Du and Guo (2011) [10]. In both cases a spreading-vanishing dichotomy was established, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or fails to establish and dies out in the long run; moreover, in the case of spreading, the asymptotic spreading speed was determined. In this paper, we consider the non-radially symmetric case. In such a situation, similar to the classical Stefan problem, smooth solutions need not exist even if the initial data are smooth. We thus introduce and study the “weak solution” for a class of free boundary problems that include the Fisher–KPP as a special case. We establish the existence and uniqueness of the weak solution, and through suitable comparison arguments, we extend some of the results obtained earlier in Du and Lin (2010) [11] and Du and Guo (2011) [10] to this general case. We also show that the classical Aronson–Weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here.