Related papers: Multi-dimensional Burgers equation with unbounded …
This paper deals with the Cauchy problem associated to the nonlinear fourth-order Schr\"odinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation $i\partial _{t}u+\epsilon \Delta u+\delta A…
Due to the highly degeneracy and singularities of the entropy equation, the physical entropy for viscous and heat conductive polytropic gases behave singularly in the presence of vacuum and it is thus a challenge to study its dynamics. It…
Within the class of nonlinear hyperbolic balance laws posed on a curved spacetime (endowed with a volume form), we identify a hyperbolic balance law that enjoys the same Lorentz invariance property as the one satisfied by the Euler…
We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth…
In this paper we are extending the well known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example we apply the theory to the case of a differential-difference…
We consider multidimensional stochastic Burgers equation on the torus $\mathbb{T}^d$ and the whole space $\Rd$. In both cases we show that for positive viscosity $\nu>0$ there exists a unique strong global solution in $L^p$ for $p>d$. In…
We consider the Cauchy problem for the generalized Kadomtsev--Petviashvili--Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropic dissipative term. Under some suitable…
In this paper, we study the Cauchy problem for the Benjamin-Ono-Burgers equation $\partial_t u-\epsilon \partial_x^2 u+\mathcal{H}\partial_x^2u+u u_x=0$, where $\mathcal{H}$ denotes the Hilbert transform. We obtain that it is uniformly…
We prove $L^2$ stability estimates for entropic shocks among weak, possibly \emph{non-entropic}, solutions of scalar conservation laws $\partial_t u+\partial_x f(u)=0$ with strictly convex flux function $f$. This generalizes previous…
The Cauchy problem for the two dimensional compressible Euler equations with data in the Sobolev space $H^s(\mathbb R^2)$ is known to have a unique solution of the same Sobolev class for a short time, and the data-to-solution map is…
In this article, we develop what are, to the best of our knowledge, the first negative results for scalar conservation laws. We begin with explicit examples where bounded initial data leads to $L^{\infty}$ blow-up despite flux regularity.…
Parabolic integro-differential model Cauchy problem is considered in the scale of Lp -spaces of functions whose regularity is defined by a scalable Levy measure. Existence and uniqueness of a solution is proved by deriving apriori…
The present paper is devoted to the study of the well-posedness issue for the density-dependent Euler equations in the whole space. We establish local-in-time results for the Cauchy problem pertaining to data in the Besov spaces embedded in…
This paper studies the global existence and uniqueness of strong solutions and its large-time behavior for the compressible isothermal Euler equations with a nonlocal dissipation. The system is rigorously derived from the kinetic…
The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in…
We study a family of equations defined on the space of tensor densities of weight $\lambda$ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the…
We prove that if $u$ is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis…
We deal with the Cauchy problem for multi-dimensional scalar conservation laws, where the fluxes and the source terms can be discontinuous functions of the unknown. The main novelty of the paper is the introduction of a~kinetic formulation…
This work is concerned about the Cauchy problem for the following generalized KdV- Burgers equation \begin{equation*} \left\{\begin{array}{l} \partial_tu+\partial_x^3u+L_pu+u\partial_xu=0, u(0,\,x)=u_0(x). \end{array} \right.…
We consider the Cauchy problem for the integrable nonlocal nonlinear Schr\"odinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2\sigma q^{2}(x,t)\overline{q(-x,t)}=0 $ with initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is…