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Given that the terminal condition is of at most linear growth, it is well known that a Cauchy problem admits a unique classical solution when the coefficient multiplying the second derivative (i.e., the volatility) is also a function of at…
We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the…
The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the…
Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds,…
In this paper, we study the Cauchy problem to the 3D fractional compressible isentropic generalized Navier-Stokes equations for viscous compressible fluid with one Levy diffusion process. We obtain the existence and uniqueness of global…
For smooth initial data, we establish the global existence and uniqueness of strong and classical solutions to the Cauchy problem for the barotropic compressible Navier-Stokes equations in two spatial dimensions with vacuum state as far…
Main purpose of this paper is to study the local propagation of singularities of viscosity solution to contact type evolutionary Hamilton-Jacobi equation $$ D_tu(t,x)+H(t,x,D_xu(t,x),u(t,x))=0. $$ An important issue of this topic is the…
This paper concerns the Cauchy problem of the barotropic compressible Navier-Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear…
We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.
The existence of solutions to Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients is considered in a space of integrable functions. First, we consider the existence and uniqueness of…
In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by an $\mathbb{F}$-semimartingale $M$ and a finite cubic variation process $\xi$ which has the structure…
We consider the Cauchy problem for the isentropic compressible Euler equations in a three-dimensional periodic domain under general pressure laws. For any smooth initial density away from the vacuum, we construct infinitely many entropy…
This paper studies the local existence of strong solutions to the Cauchy problem of the 2D fluid-particle interaction model with vacuum as far field density. Notice that the technique used by Ding et al.\cite{SBH} for the corresponding 3D…
In this paper, we prove that the existence and uniqueness of globally weak solutions to the Cauchy problem for the weakly dissipative Camassa-Holm equation in time weighted $H^1$ space. First, we derive an equivalent semi-linear system by…
In this paper we investigate the well-posedness of the Cauchy problem for a Schr\"odinger operator with singular lower order terms. We allow distributional coefficients and we approach this problem via the regularising methods at the core…
We use the contracting mapping principle for proving that under some mild restrictions the Cauchy problem for quasilinear systems of functional differential equations with retarded arguments has the unique solution. As a consequence from…
In this paper, we investigate the Cauchy problem of the compressible non-resistive MHD on $\mathbb{R}^2$ with vacuum as far field density. We prove that the 2D Cauchy problem has a unique local strong solution provided the initial density…
In this note we propose a definition of weak solution for an abstract Cauchy problem in a Hilbert space, and we discuss existence and uniqueness results.
In this paper, we present an analytical study, in the one space dimensional case, of the fluid dynamics system proposed in [4] to model the formation of biofilms. After showing the hyperbolicity of the system, we show that, in a open…
In Part I of the paper, we prove non-uniqueness of the solution to the Cauchy problem of the Euler equations of an ideal incompressible fluid in dimension two with vorticity in some Lebesgue space. The radially symmetric external force is…