Blow-Up of Solutions to Fractional-in-Space Burgers-Type Equations

We consider fractional-in-space analogues of Burgers equation and Korteweg-de Vries-Burgers equation on bounded domains. Namely, we establish sufficient conditions for finite-time blow-up of solutions to the mentioned equations. The obtained conditions depend on the initial value and the boundary conditions. Some examples are provided to illustrate our obtained results. In the proofs of our main results, we make use of the test function method and some integral inequalities.

Self-adjointness and conservation laws of Burgers-type equations

This paper focuses on the self-adjointness of some Burgers-type equations based on the existing definitions. It follows that the corresponding Frobenius Burgers-type equations are constructed by taking values in a commutative subalgebra [Formula: see text]. In order to investigate the self-adjointness of these Frobenius-type equations, we introduce a few additional notations and definitions. Additionally, the conservation laws are obtained of several equations studied by using symmetries.

Stochastic Entropy Solutions for Stochastic Scalar Balance Laws

We are concerned with the initial value problem for a multidimensional balance law with multiplicative stochastic perturbations of Brownian type. Using the stochastic kinetic formulation and the Bhatnagar-Gross-Krook approximation, we prove the uniqueness and existence of stochastic entropy solutions. Furthermore, as applications, we derive the uniqueness and existence of the stochastic entropy solution for stochastic Buckley-Leverett equations and generalized stochastic Burgers type equations.